Universal Bounds for Size and Energy of Codes of Given Minimum and Maximum Distances
Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, Maya Stoyanova
aa r X i v : . [ c s . I T ] O c t Universal Bounds for Size and Energy of Codes of GivenMinimum and Maximum Distances
October 17, 2019
P. Boyvalenkov [email protected]
Institute for Mathematics and Informatics, BAS, Sofia 1113, Bulgariaand Southwestern University, Blagoevgrad, Bulgaria
P. Dragnev [email protected]
Department of Mathematical Sciences, PFW, Fort Wayne, IN 46805, USA
D. Hardin [email protected]
Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA
E. Saff [email protected]
Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA
M. Stoyanova [email protected]
Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, Sofia 1164, Bulgaria
Abstract
We employ signed measures that are positive definite up to certain degrees to establishLevenshtein-type upper bounds on the cardinality of codes with given minimum and maximumdistances, and universal lower bounds on the potential energy (for absolutely monotone inter-actions) for codes with given maximum distance and cardinality. The distance distributionsof codes that attain the bounds are found in terms of the parameters of Levenshtein-typequadrature formulas. Necessary and sufficient conditions for the optimality of our boundsare derived. Further, we obtain upper bounds on the energy of codes of fixed minimum andmaximum distances and cardinality.
Keywords —bounds for codes, linear programming, energy of codes.
Let F q be an alphabet of size q . We consider codes (sets) C ⊂ F nq = { ( x , . . . , x n ) : x i ∈ F q } withthe Hamming distance d ( x, y ) between words x, y ∈ F nq . In setting of F nq as a polynomial metricspace [26] the following change of the variable t = 1 − dn ∈ T n := (cid:26) t i = − in : i = 0 , , . . . , n (cid:27)
1s very convenient. It brings the distances to "inner" products and for x, y ∈ F nq we write h x, y i = 1 − d ( x, y ) n = t n − d ∈ T n . For any code C ⊂ F nq we use s ( C ) := max {h x, y i : x, y ∈ C, x = y } ∈ T n ,ℓ ( C ) := min {h x, y i : x, y ∈ C, x = y } ∈ T n , to denote the counterparts of the minimum and maximum distance of C , respectively. Denote by C n,q ( ℓ, s ) := { C ⊂ F nq | s ( C ) ≤ s, ℓ ( C ) ≥ ℓ } (1)the set of codes in F nq with pairwise distances greater than or equal to the minimum distance d := n (1 − s ) / and less than or equal to the maximum distance D := n (1 − ℓ ) / . Let A q ( n, ℓ, s ) := max {| C | : C ∈ C n,q ( ℓ, s ) } be the maximum possible cardinality of a code from C n,q ( ℓ, s ) . The investigation of the quantitieslike A q ( n, ℓ, s ) is one of the classical problems in the coding theory.We are interested also in a minimum energy problem which is somewhat more general butturns out to be closely related. Definition 1.1
Given a (potential) function h ( t ) : [ − , → [0 , + ∞ ] and a code C ⊂ F nq , the potential energy (also referred to as h -energy ) of C is E h ( C ) := X x,y ∈ C,x = y h ( h x, y i ) . While we only need the values of h on the discrete set T n for computing the h -energy, wefurther assume that h is (strictly) absolutely monotone on the interval [-1,1); that is, h and all itsderivatives are defined and (positive) nonnegative on this interval. This approach facilitates ourinvestigation and the explanation of our results. We remark that the function F ( z ) = h ( t ) , where z = n (1 − t ) / , is completely monotone on (0 , n ] (i.e., ( − k F ( k ) ( z ) ≥ for all z ∈ (0 , n ] ) if andonly if h is absolutely monotone on [ − , .For absolutely monotone potentials h we consider the quantity E h ( n, M, ℓ ) := min { E h ( C ) : C ∈ C n,q ( ℓ, − /n ) , | C | = M } , the smallest possible h -energy of a code from C n,q ( ℓ, − /n ) with prescribed M .General linear programming bounds for quantities like A q ( n, ℓ, s ) and E h ( n, M, ℓ ) were first in-troduced by Delsarte [17] (see [18,26] and references therein) and Yudin [29]. Linear programmingbounds for energies of codes and designs in different spaces (including F nq ) were investigated forthe first time by Ashikhmin-Barg [1], Ashikhmin-Barg-Litsyn [2] (see also [3, 4] Energies of codesin F nq were considered in 2014 by Cohn and Zhao [15] (see also [14]) with a focus on (universally)optimal codes and by the authors [9] who focused on universal bounds.2n this paper we use linear programming techniques to derive explicit upper bounds for A q ( n, ℓ, s ) and lower bounds for E h ( n, M, ℓ ) . Our bounds can be computed for all feasible valuesof q , n , s , and ℓ , which makes them universal in the sense of Levenshtein [26]. We are not aware ofsuch explicit universal bounds in the existing literature (see [21] for a particular case) more than20 years after the chapter [26] by Levenshtein and the paper [18] by Delsarte and Levenshtein.There is an intricate interplay between the Levenshtein universal bounds for A q ( n, − , s ) anduniversal lower bounds on E h ( n, M, − in different polynomial metric spaces (see [8] for Euclideanspheres S n − and [9] for Hamming spaces F nq ). We further that relationship to correspondingbounds for codes from C n,q ( ℓ, − /n ) to derive and investigate simultaneously our cardinalityand energy bounds.For any real polynomial f ( t ) we consider its expansion in Krawtchouk polynomials (see Sec-tion 2), f ( t ) = n X i =0 f i Q ( n,q ) i ( t ) (if the degree of the polynomial f ( t ) exceeds n , then f ( t ) is taken modulo Q ni =0 ( t − t i ) ) and set F ≥ := { f ( t ) : f > , f i ≥ , i = 1 , , . . . , n } . If f i > for i = 0 , , , . . . , deg( f ) , then we write f ( t ) ∈ F > . The coefficient f is of specialinterest and we call it the zeroth coefficient of f ( t ) .Following Delsarte [17], we have A q ( n, ℓ, s ) ≤ min f ∈F n,ℓ,s f (1) f , (2)where F n,ℓ,s := { f ∈ F ≥ : f ( t ) ≤ , t ∈ [ ℓ, s ] } . Similarly, following Yudin [29], we have E h ( n, M, ℓ ) ≥ max g ∈G ( h ) n,ℓ M ( M g − g (1)) , (3)where G ( h ) n,ℓ := { g ∈ F ≥ : g ( t ) ≤ h ( t ) , t ∈ [ ℓ, } . Therefore, major results in this context crucially depend on proper choice and investigation ofpolynomials that optimize (2) or (3).The Levenshtein bound (see [24–26]) and the energy bound [9] work for ℓ = − and, ofcourse, depend on the properties of Krawtchouk polynomials and their adjacent polynomials whichare orthogonal with respect to classical positive measures. The case ℓ > − , however, alreadyinvolves more challenging signed measures. In this paper we develop the necessary theory of signedmeasures to be used in the investigation of the optimization problems arising from the right handsides of (2) and (3). Then we derive and investigate universal upper bounds for A q ( n, ℓ, s ) andlower bounds for E h ( n, M, ℓ ) . 3he paper is organized as follows. In Sections 2 and 3 we introduce the so-called adjacentpolynomials and signed measures. Then we establish the positive definiteness of the correspondingmeasures up to appropriate degrees. Properties of the associated orthogonal (and adjacent again)polynomials are derived and discussed in Section 4, where we define Levenshtein-type polynomials f n,ℓ,s k ( t ) to be used in (2). A Levenshtein-type quadrature formula is derived with nodes the rootsof f n,ℓ,s k ( t ) to serve in proofs and properties. In Section 5 we obtain simultaneously Levenshtein-type upper bounds on A q ( n, ℓ, s ) and (as in the case ℓ = − ) the strongly related lower boundson E h ( n, M, ℓ ) . An important role in the proof is played by what we call the ( k, ℓ ) -strengthenedKrein condition extending the Levenshtein’s strengthened Krein condition. Section 6 is devotedto description of codes which would attain our bounds. The distance distributions of such codesare found as functions of corresponding quadrature formulas parameters. In Section 7 we provenecessary and sufficient conditions for the optimality of our bounds (in other words, for existenceof improving polynomials from the sets F n,ℓ,s and G ( h ) n,ℓ ). The optimality (or existence of improve-ments) happens only simultaneously for both bounds. A linear programming refinement of ourbounds is discussed in Section 8, where we provide evidence that in most cases the the nodesof our polynomials serve as the best approximation for the general linear programming solution.Upper bounds on the energy of the codes from C n,q ( ℓ, s ) (including the case ℓ = − ) of fixedcardinality M are derived in Section 9 providing this way a strip where the energies of all suchcodes belong. Examples are shown in Section 10 where we build a Levenshtein-type system ofbounds for a fixed ℓ . For fixed n and q , the (normalized) Krawtchouk polynomials are defined by Q ( n,q ) i ( t ) := 1 r i K ( n,q ) i ( z ) , where z = n (1 − t )2 is a change of the variable between the set { , , . . . , n } of the distances in F nq and the set T n , r i := ( q − i (cid:18) ni (cid:19) , i = 0 , , . . . , n, are certain dimensions of spaces of functions (see also (5) below), and K ( n,q ) i ( z ) := i X j =0 ( − j ( q − i − j (cid:18) zj (cid:19)(cid:18) n − zi − j (cid:19) ,i = 0 , , . . . , n , are the (usual) Krawtchouk polynomials corresponding to F nq (see [28, Section2.82]). In the sequel we will omit the index ( n, q ) in the notation of Krawtchouk polynomials.4he polynomials { Q i ( t ) } ni =0 form a basis of the space P n of real polynomials of degree at most n and satisfy the following three-term recurrence relation ( t − a i ) Q i ( t ) = b i Q i +1 ( t ) + c i Q i − ( t ) ,i = 1 , , . . . , n − , with initial conditions Q ( t ) = 1 and Q ( t ) = ( qt + q − / (2( q − , where a i = − ( q − n − i ) qn ,b i = 2( q − n − i ) qn , c i = 2 iqn . The measure of orthogonality for the system { Q i ( t ) } ni =0 is discrete and given by µ n := q − n n X i =0 r n − i δ t i , (4)where δ t i is the Dirac-delta measure at t i ∈ T n . The form h f, g i = Z f ( t ) g ( t ) dµ n ( t ) defines an inner product over the class of polynomials of degree at most n . Note that r i = (cid:18)Z − ( Q i ( t )) dµ n ( t ) (cid:19) − = k Q i k − . (5)We also need (1 , and (1 , adjacent polynomials as introduced by Levenshtein (cf. [26,Section 6.2], see also [24, 25]). Denote T i ( x, y ) := i X j =0 r j Q j ( x ) Q j ( y ) , (6)and define [26, Eq. (5.65)] Q , i ( t ) := T i ( t, T i (1 , , i = 0 , , . . . , n − . (7)Similarly, denote T , i ( x, y ) := i X j =0 r , j Q , j ( x ) Q , j ( y ) , (8)where r , j = (cid:16)P ju =0 r u (cid:17) (cid:0) n − j (cid:1) ( q − j , j = 0 , , . . . , n − , (1 , counterparts of r j , and define [26, Eq. (5.68)] Q , i ( t ) := T , i ( t, − T , i (1 , − , i = 0 , , . . . , n − . (9)Note that r , j = (cid:16)P ju =0 (cid:0) n − u (cid:1) ( q − u (cid:17) (cid:0) n − j (cid:1) ( q − j , j = 0 , , . . . , n − , give the explicit formulas for the (1 , norm of the polynomials Q , j ( t ) similatly to (5).The corresponding measures of orthogonality of the systems { Q , i ( t ) } n − i =0 and { Q , i ( t ) } n − i =0 are c , (1 − t ) dµ n ( t ) , c , (1 − t )(1 + t ) dµ n ( t ) , (10)respectively, where c , = q q − , c , = nq n − q − are normalizing constants (see [26, Section 6.2]). Of course, the adjacent polynomials also satisfycorresponding three-term recurrence relations ( t − a ,εi ) Q ,εi ( t ) = b ,εi Q ,ℓi +1 ( t ) + c ,εi Q ,εi − ( t ) , where ε ∈ { , } , b ,εi > is the ratio of the leading coefficients of Q ,εi +1 ( t ) and Q ,εi ( t ) , c ,εi = r ,εi − b ,εi − /r ,εi > and a ,εi = 1 − b ,εi − c ,εi .Note also the explicit relations [25] Q , i ( t ) = K ( n − ,q ) i ( z − P ij =0 r j ,Q , i ( t ) = K ( n − ,q ) i ( z − P ij =0 (cid:0) n − j (cid:1) ( q − j , where z = n (1 − t ) / as above, between the (1 , and (1 , adjacent polynomials and the usualKrawtchouk polynomials.For ℓ ∈ T n we shall introduce below further adjacent polynomials Q ,ℓi ( t ) as generalizationsof Q , i ( t ) (note that ℓ = − in Q ,ℓi ( t ) gives Q , i ( t ) by the definitions in [26, Eqn. (5.66)] and(21) below). Under certain natural conditions the polynomials Q ,ℓi ( t ) are orthogonal with respectto a signed measure dµ n,ℓ ( t ) which is defined and investigated below. With the next step, weshall use this new series to construct polynomials Q ,ℓ,si ( t ) which are orthogonal with respect toanother signed measure dµ n,ℓ,s ( t ) again to be defined and investigated below. Furthermore, thesigned measures dµ n,ℓ ( t ) and dµ n,ℓ,s ( t ) are strong enough to imply properties which are crucialfor our constructions. Then our Levenshtein-type polynomials will be constructed to be appliedin (2) and, moreover, as in the case ℓ = − [9], to setup polynomials to be applied in (3). In all6hese constructions and investigations, the Christoffel-Darboux formula [28, Chapter 3.2] plays animportant role.These Levenshtein-type polynomials can also be viewed as adjacent polynomials summarizedby the following sequence: Q i → Q , i → Q ,ℓi → Q ,ℓ,si , (11)where each subsequent family of polynomials can be expressed in terms of the previous familyusing the Christoffel-Darboux formula (see (7), (9), (21), and (25)).We conclude this section with notations for the zeros of the polynomials from the sequence(11). Let t ai, < t ai, < · · · < t ai,i be the zeros of the polynomial Q ai ( t ) , i = 0 , , . . . , , where the index a stands for the pairs (1 , , (1 , , (1 , ℓ ) , or the triple (1 , ℓ, s ) , respectively. Signed measures were first used by Cohn and Kumar in [14] in the context of linear programmingbounds for energy of spherical codes.
Definition 3.1
A signed Borel measure µ on R for which all polynomials are integrable is calledpositive definite up to degree m if for all real polynomials p of degree at most m we have R p ( t ) dµ ( t ) > . For such µ , the bi-linear form h f, g i µ := Z f ( t ) g ( t ) dµ ( t ) , (12) is an inner product on the space P m . Suppose that k , ℓ , and s are such that the roots of Q , k ( t ) lie in the open interval ( ℓ, s ) ⊂ ( − , ; that is, (note ℓ < ) − ≤ ℓ < t , k, < t , k,k < s ≤ . (13)Then we define the following signed measures on [ − , (see (4) and (10)) dµ n,ℓ ( t ) := c ,ℓ ( t − ℓ )(1 − t ) dµ n ( t ) , (14) dµ n,s ( t ) := c ,s ( s − t )(1 − t ) dµ n ( t ) , (15) dµ n,ℓ,s ( t ) := c ,ℓ,s ( t − ℓ )( s − t )(1 − t ) dµ n ( t ) . (16)The normalizing constants in (14)–(16) are given by c ,ℓ := nq q − n − − nq (1 + ℓ )) ,c ,s := nq q − nq (1 + s ) − n − , ,ℓ,s := n q q − n − nqd − n − q + 2) − n q d )] , where d = (2 + ℓ + s ) / and d = (1 + ℓ )(1 + s ) . We will show below that for n , q , ℓ and s thatsatisfy (13) the constants c ,ℓ , c ,s , and c ,ℓ,s are all positive.The following theorem establishes the positive definiteness of the signed measures (14)–(16)up to degrees k − , k − , and k − , respectively, as a consequence of the appropriate location(13) of ℓ and s . Theorem 3.2
For given positive integers n ≥ and q ≥ , let k , ℓ , and s satisfy the inequalities (13) . Then the measures µ n,ℓ , µ n,s and µ n,ℓ,s are positive definite up to degrees k − , k − , and k − , respectively.Proof. Modifying the classical Radau quadrature [16, Sec. 2.7] for integration with respect todiscrete measures we conclude that the zeros of the corresponding discrete orthogonal polynomial,the system of k + 1 nodes t , k, < t , k, < · · · < t , k,k < defines a positive (i.e., the weights w i , i = 1 , , . . . , k + 1 , are positive) Radau quadrature withrespect to µ n , f := Z − f ( t ) dµ n ( t ) = w k +1 f (1) + k X i =1 w i f ( t , k,i ) , (17)that is exact for all polynomials of degree at most k .Using (17) for f ( t ) = ( t − ℓ )(1 − t ) , we find that (cid:16) c ,ℓ (cid:17) − = k X i =1 w i ( t , k,i − ℓ )(1 − t , k,i ) > . Similarly, we can show that c ,s > and c ,ℓ,s > .Next, we apply (17) for q ( t ) , an arbitrary polynomial of degree at most k − , to see that Z − q ( t ) dµ n,ℓ ( t )= c ,ℓ Z − q ( t )( t − ℓ )(1 − t ) dµ n ( t )= c ,ℓ k X i =1 w i q ( t , k,i )( t , k,i − ℓ )(1 − t , k,i ) ≥ . The equality holds only if q ( t , k,i ) = 0 for all i = 1 , . . . , k , which would imply that q ( t ) ≡ .Therefore the measure dµ n,ℓ ( t ) is positive definite up to degree k − . That µ n,s is positive definiteup to degree k − provided s > t , k,k follows similarly.8inally, if q ( t ) has degree at most k − , then we utilize (17) again to see that Z − q ( t ) dµ n,ℓ,s ( t )= c ,ℓ,s Z − q ( t )( t − ℓ )( s − t )(1 − t ) dµ n ( t )= c ,ℓ,s k X i =1 w i q ( t , k,i )( t , k,i − ℓ )( s − t , k )(1 − t , k,i ) > . This implies that the measure dµ n,ℓ,s ( t ) is positive definite up to degree k − , as required. (cid:3) Theorem 3.2 allows us to define orthogonal polynomials with respect to the correspondingsigned measures. This provides essential ingredients for modifying Levenshtein’s framework. Q ,ℓj ( t ) , j = 0 , , . . . , k , and Q ,ℓ,sj ( t ) , j = 0 , , . . . , k − Some of the basic properties of orthogonal polynomials are no longer valid for series of polyno-mials generated by signed measures. Fortunately, our measures dµ n,ℓ ( t ) and dµ n,ℓ,s ( t ) possessthe necessary properties by Theorem 3.2. Applying Gram-Schmidt orthogonalization we derivethe existence and uniqueness (for the so-chosen normalizations) of the following two classes oforthogonal polynomials thus completing the sequence (11). Theorem 4.1
For given positive integers n ≥ , q ≥ , let k , ℓ , and s satisfy the inequalities (13) . The following two classes of orthogonal polynomials are well-defined: { Q ,ℓj ( t ) } kj =0 , w . r . t . dµ n,ℓ ( t ) , Q ,ℓj (1) = 1; { Q ,ℓ,sj ( t ) } k − j =0 , w . r . t . dµ n,ℓ,s ( t ) , Q ,ℓ,sj (1) = 1 . The polynomials in both classes satisfy a three-term recurrence relation and their zeros interlace.
For our purposes we shall restrict to values of ℓ such that Q , k +1 ( ℓ ) Q , k ( ℓ ) < . (18)As shown in the proof of Theorem 4.2 below the condition (18) is equivalent to the requirementfor the largest zero of Q ,ℓk ( t ) to be less than . 9 .2 Explicit construction and investigation of the polynomials Q ,ℓj ( t ) , j =0 , , . . . , k The explicit form of the polynomials Q ,ℓj ( t ) can be seen as a straightforward generalization of (6)-(7) by using ℓ instead of − . We utilize the Christoffel-Darboux formula (see, for example [28, Th.3.2.2], [26, Eq. (5.65)]) T , i ( x, y ) r , i b , i = Q , i +1 ( x ) Q , i ( y ) − Q , i +1 ( y ) Q , i ( x ) x − y (19)(when x = y appropriate derivatives are used) in our construction. Moreover, similarly to [26],we use (19) to prove the interlacing properties of the zeros of { Q ,ℓi ( t ) } with respect to the zerosof { Q , i ( t ) } .In what follows in this and the next sections we assume that t , k +1 , < ℓ < t , k, . (20) Theorem 4.2
Let n , q , k , and ℓ be such that (18) and (20) are satisfied. Then Q ,ℓi ( t ) = T , i ( t, ℓ ) T , i (1 , ℓ ) = η ,ℓi t i + · · · , i = 0 , , . . . , k, (21) with η ,ℓi > and the polynomial Q ,ℓi ( t ) has i simple zeros t ,ℓi, < t ,ℓi, < · · · < t ,ℓi,i in the interval ( ℓ, . Furthermore, the following interlacing properties t ,ℓi,j ∈ (cid:16) t , i,j , t , i +1 ,j +1 (cid:17) , i = 1 , . . . , k − , j = 1 , . . . , i ; t ,ℓk,j ∈ (cid:16) t , k +1 ,j +1 , t , k,j +1 (cid:17) , j = 1 , . . . , k − , (22) and, finally, t ,ℓk,k ∈ (cid:16) t , k +1 ,k +1 , (cid:17) hold true.Proof. It follows from (19) that the kernel T , i ( t, ℓ ) is orthogonal to any polynomial of degreeat most i − with respect to the measure µ n,ℓ ( t ) . Hence (21) follows from the positive definitenessof dµ n,ℓ ( t ) up to degree k − and the uniqueness of the Gram-Schmidt orthogonalization process(note also the normalization). The comparison of coefficients in (21) yields η ,ℓi > , i = 0 , , . . . , k .Next, it follows from (19) and (21) that the solutions of the equation Q , i +1 ( t ) Q , i ( t ) = Q , i +1 ( ℓ ) Q , i ( ℓ ) (23)are the zeros of Q ,ℓi ( t ) and the number ℓ .For every i < k the zeros of Q , i +1 ( t ) and Q , i ( t ) are interlaced and contained in the interval h t , k, , t , k,k i . Since sign Q , i ( ℓ ) = ( − i , we have Q , i +1 ( ℓ ) /Q , i ( ℓ ) < . The rational function Q , i +1 ( t ) /Q , i ( t ) has simple poles at t , i,j , j = 1 , . . . , i , and simple zeros at t , i +1 ,j , j = 1 , . . . , i + 1 .10herefore, there is at least one solution t ,ℓi,j of (23) on each interval (cid:16) t , i,j , t , i +1 ,j +1 (cid:17) , j = 1 , . . . , i ,which accounts exactly for the zeros of Q ,ℓi ( t ) .When i = k we have Q , k +1 ( ℓ ) /Q , k ( ℓ ) > . Since ℓ ∈ (cid:16) t , k +1 , , t , k, (cid:17) , we account similarly forthe first k − solutions of (23), namely t ,ℓk,j ∈ (cid:16) t , k +1 ,j +1 , t , k,j +1 (cid:17) , j = 1 , . . . , k − , to establish theinterlacing properties (22). For the last zero of Q ,ℓk ( t ) we use the fact that Q , k +1 ( t ) /Q , k ( t ) > for t ∈ ( t , k +1 ,k +1 , ∞ ) . As lim t →∞ Q , k +1 ( t ) /Q , k ( t ) = ∞ , we have one more solution t ,ℓk,k > t , k +1 ,k +1 of (23). Then (18) implies that t ,ℓk,k < because Q , k +1 (1) /Q , k (1) = 1 . (cid:3) The positive definiteness of the measure dµ n,ℓ ( t ) implies that r ,ℓi := (cid:18)Z − (cid:16) Q ,ℓi ( t ) (cid:17) dµ n,ℓ ( t ) (cid:19) − > for i = 0 , , . . . , k − . The three-term recurrence relation from Theorem 4.1 can be written as ( t − a ,ℓi ) Q ,ℓi ( t ) = b ,ℓi Q ,ℓi +1 ( t ) + c ,ℓi Q ,ℓi − ( t ) ,i = 1 , , . . . , k − , where b ,ℓi = η ,ℓi +1 η ,ℓi > , c ,ℓi = r ,ℓi − b ,ℓi − r ,ℓi > , a ,ℓi = 1 − b ,ℓi − c ,ℓi . The initial conditions are Q ,ℓ ( t ) = 1 and Q ,ℓ ( t ) = nq ( nqℓ + nq − n + 2) t + A B , where A = n ( q − qℓ + q −
2) + n ( qℓ + 5 q − − q − ,B = n ( q − qℓ + q −
2) + 2 n ( qℓ + 4 q − − q − . Finally in this description we note that by Theorem 4.1 the zeros of the polynomials Q ,ℓj ( t ) interlace; i.e., t ,ℓj,i < t ,ℓj − ,i < t ,ℓj,i +1 , i = 1 , , . . . , j − . We conclude this section with a property of the polynomials Q ,ℓi ( t ) which will give a particularanswer to positive definiteness problems to arise below. Lemma 4.3 If ℓ is as in (13) , then ( t − ℓ ) Q ,ℓi ( t ) ∈ F > for every i = 0 , , . . . , k − .Proof. For every i = 0 , , . . . , k − it follows from (19) and (21) that ( t − ℓ ) Q ,ℓi ( t ) = 1 − ℓ − q i (cid:16) Q , i +1 ( t ) − q i Q , i ( t ) (cid:17) , where q i = Q , i +1 ( ℓ ) /Q , i ( ℓ ) < as in the proof of Theorem 4.2. Now Q , j ( t ) ∈ F > (this isimmediate from the definitions (6)-(7)) completes the proof. (cid:3) .3 Construction and investigation of Q ,ℓ,sk − ( t ) and Levenshtein-type polynomi-als In this section we perform the next step in our construction. Using the system { Q ,ℓi ( t ) } ki =0 fromthe previous section we derive explicitly polynomials Q ,ℓ,si ( t ) , i = 1 , , . . . , k − , orthogonal withrespect to the measure µ n,ℓ,s ( t ) . The last polynomial in this sequence, Q ,ℓ,sk − ( t ) , will be the mainingredient in our Levenshtein-type polynomials.Consider the Christoffel-Darboux kernel associated with the polynomials Q ,ℓj ( t ) : R ,ℓi ( x, y ) := i X j =0 r ,ℓj Q ,ℓj ( x ) Q ,ℓj ( y )= r ,ℓi b ,ℓi Q ,ℓi +1 ( x ) Q ,ℓi ( y ) − Q ,ℓi +1 ( y ) Q ,ℓi ( x ) x − y , for ≤ i ≤ k − (when x = y appropriate derivatives are used). Given (20) and assuming that t , k,k < s < t ,ℓk,k , (24)we define Q ,ℓ,si ( t ) := R ,ℓi ( t, s ) R ,ℓi (1 , s ) , i = 0 , , . . . , k − . (25)We focus on the polynomials Q ,ℓ,sk − ( t ) . Their existence, uniqueness, and the correctness ofthe definition (25) follow as in Theorem 4.2. The proof of the next assertion about the zeros of Q ,ℓ,sk − ( t ) is similar to the corresponding part of Theorem 4.2 but we include it for convenience ofthe reader. In addition to (18) we require Q ,ℓk ( s ) Q ,ℓk − ( s ) > Q ,ℓk ( ℓ ) Q ,ℓk − ( ℓ ) (26)in order to get the smallest zero of Q ,ℓ,sk − ( t ) in the interval ( ℓ, t ,ℓk, ) . Theorem 4.4
Let n , q , ℓ , s , and k be such that (20) , (24) , (18) and (26) are fulfilled. Then thepolynomial Q ,ℓ,sk − ( t ) has k − simple zeros α < α < · · · < α k − such that α ∈ ( ℓ, t ,ℓk, ) , α i +1 ∈ ( t ,ℓk − ,i , t ,ℓk,i +1 ) ,i = 1 , , . . . , k − . In particular, ℓ < α and α k − < s .Proof. It follows from the Christoffel-Darboux formula for R ,ℓi and the definition (25) thatthe solutions of the equation Q ,ℓk ( t ) Q ,ℓk − ( t ) = Q ,ℓk ( s ) Q ,ℓk − ( s ) (27)12re the zeros of Q ,ℓ,sk − ( t ) and the number s .The rational function Q ,ℓk ( t ) /Q ,ℓk − ( t ) has k − simple poles at the zeros t ,ℓk − ,i , i = 1 , , . . . , k − , of Q ,ℓk − ( t ) , and k zeros at the zeros t ,ℓk,i , i = 1 , , . . . , k , of Q ,ℓk ( t ) . Therefore, there is a solutionof (27); i.e., a zero of Q ,ℓ,sk − ( t ) , in each interval ( t ,ℓk − ,i , t ,ℓk,i +1 ) , i = 1 , , . . . , k − , which accountsexactly for k − zeros, say α < α < · · · < α k − . Moreover, since Q ,ℓk ( s ) /Q ,ℓk − ( s ) < underthe assumptions for s , we actually have α i +1 ∈ ( t ,ℓk − ,i , t ,ℓk,i +1 ) , i = 2 , , . . . , k − . Note that α k − < t ,ℓk,k − < t , k,k < s .Since the function Q ,ℓk ( t ) /Q ,ℓk − ( t ) increases from −∞ to + ∞ in the interval [ −∞ , t ,ℓk − , ) , theinequalities > Q ,ℓk ( s ) /Q ,ℓk − ( s ) > Q ,ℓk ( ℓ ) /Q ,ℓk − ( ℓ ) (see (26)) imply that the smallest zero α of Q ,ℓ,sk − ( t ) lies in the interval ( ℓ, t ,ℓk, ) .Finally, using again that Q ,ℓk ( s ) /Q ,ℓk − ( s ) < and the fact that the function Q ,ℓk ( t ) /Q ,ℓk − ( t ) strictly increases from −∞ to 1 for t ∈ ( t ,ℓk − ,k − , , we have the root s of (27) in this interval. (cid:3) We can already define the Levenshtein-type polynomial f n,ℓ,s k ( t ) := ( t − ℓ )( t − s ) (cid:16) Q ,ℓ,sk − ( t ) (cid:17) (28)and proceed with an investigation of its properties.Denote by L i ( t ) , i = 0 , , . . . , k + 1 , the Lagrange basic polynomials generated by the nodes α < α < · · · < α k − < α k < and define ρ i := Z − L i ( t ) dµ n ( t ) , i = 0 , , . . . , k + 1 . (29)The next statement is an analog of one of the main theorems (Theorem 5.39) from [26]. Itinvolves the zeros of f n,ℓ,s k ( t ) to form a right end-point Radau quadrature formula with positiveweights. Theorem 4.5
In the context of Theorem 4.4 let α := ℓ and α k := s . Then the Radau quadratureformula f = Z − f ( t ) dµ n ( t ) = ρ k +1 f (1) + k X i =0 ρ i f ( α i ) (30) is exact for all polynomials of degree at most k . Moreover, the weights ρ i , i = 0 , . . . , k , arepositive, and ρ k +1 > provided ( t − ℓ ) Q ,ℓk ( t ) ∈ F > .Proof. It follows from (29) that the formula (30) is exact for the Lagrange basis and hence forall polynomials of degree at most k + 1 . By a polynomial division, any polynomial f ( t ) of degreeat most k can be written as f ( t ) = q ( t )( t − ℓ )( t − s )(1 − t ) Q ,ℓ,sk − ( t ) + g ( t ) , deg( q ) ≤ k − and deg( g ) ≤ k + 1 . Then the orthogonality of Q ,ℓ,sk − ( t ) to all polynomials ofdegree at most k − with respect to the measure dµ n,ℓ,s ( t ) and the fact that the right-hand sideof (30) is the same for f ( t ) and g ( t ) show the exactness of the quadrature formula (30) for f ( t ) .We next employ the quadrature formula (30) to show the positivity of its weights ρ i .Using the polynomial f ( t ) = (1 − t )( t − ℓ ) (cid:16) Q ,ℓ,sk − ( t ) (cid:17) in (30) we obtain ρ k f ( s ) = Z − (cid:16) Q ,ℓ,sk − ( t ) (cid:17) dµ n,ℓ ( t ) > because of the positive definiteness (up to degree k − ) of the measure dµ n,ℓ ( t ) . Now f ( s ) > implies ρ k > . Similarly, with the polynomial f ( t ) = (1 − t )( s − t ) (cid:16) Q ,ℓ,sk − ( t ) (cid:17) in (30) and thepositive definiteness (up to degree k − ) of the measure dµ n,s ( t ) (see Theorem 3.2) we concludethat ρ > .To see that ρ i > for i = 1 , , . . . , k − , we use the polynomials f ( t ) = (1 − t )( t − ℓ )( s − t ) u k − ,i ( t ) , respectively, in (30), where u k − ,i ( t ) = Q ,ℓ,sk − ( t ) / ( t − α i ) . Then deg( u k − ,i ) = k − and the positive definiteness of dµ n,ℓ,s ( t ) (up to degree k − ) yields ρ i f ( α i ) = Z − u k − ,i ( t ) dµ n,ℓ,s ( t ) > . Since f ( α i ) > for this choice of f , we derive that ρ i > .Finally, we consider the weight ρ k +1 . In this case we use f ( t ) = f n,ℓ,s k ( t ) in (30) and find that f = ρ k +1 f (1) = ρ k +1 (1 − s )(1 − ℓ ) . Thus it is enough to see that the zeroth coefficient of f n,ℓ,s k ( t ) is positive. We use (25) to obtainthat f is equal to R − ( t − ℓ )( s − t )(1 − t ) Q ,ℓ,sk − ( t ) Q ,ℓ,sk − ( t ) − Q ,ℓ,sk − (1) t − dµ n ( t )+ R − ( t − ℓ )( t − s ) Q ,ℓ,sk − ( t ) dµ n ( t ) (31) = − s − p k R − ( t − ℓ ) (cid:16) Q ,ℓk ( t ) − p k Q ,ℓk − ( t ) (cid:17) dµ n ( t ) , where p k = Q ,ℓk ( s ) /Q ,ℓk − ( s ) < . Then, under the assumption ( t − ℓ ) Q ,ℓk ( t ) ∈ F > and with ( t − ℓ ) Q ,ℓk − ( t ) ∈ F > from Lemma 4.3, it follows that the last integrand belongs to F > and inparticular its zeroth coefficient is positive. This completes the proof of the theorem. (cid:3) Remark 4.6
The polynomials f n,ℓ,s k ( t ) can be also constructed via the system { Q ,si ( t ) } ki =0 insteadof { Q ,ℓi ( t ) } ki =0 in the sequence (11) . Of course, the resulting system { Q n,ℓ,si ( t ) } k − i =0 is the same. In the proof of the positive definiteness of his polynomials Levenshtein used (see [26, (3.88) and(3.92)]) what he called the strengthened Krein condition ( t + 1) Q , i ( t ) Q , j ( t ) ∈ F > (32)14or every i, j ∈ { , , . . . , n − } . We need the following modification. Definition 5.1
We say that the polynomials { Q ,ℓi ( t ) } ki =0 satisfy ( k, ℓ ) -strengthened Krein condi-tion if ( t − ℓ ) Q ,ℓi ( t ) Q ,ℓj ( t ) ∈ F > (33) for every i, j ∈ { , , . . . , k } except possibly for i = j = k . The strengthened Krein condition (32) holds true in F nq for all admissible i and j (see [26,Lemma 3.25]). However, the ( k, ℓ ) -strengthened Krein condition (33) is not true for every ℓ , andfor fixed ℓ is true only for relatively small k . On the other hand, for fixed n , all relevant pairs ( k, ℓ ) are finitely many and can be therefore subject to computational checks. Lemma 4.3 saysthat the condition is satisfied for all pairs ( i, , i = 0 , , . . . , k − .The main result in this paper is the following. It includes our Levenshtein-type upper boundon A q ( n, ℓ, s ) as an analog of Theorem 5.42 of [26] and its counterpart, a universal lower boundon E h ( n, M, ℓ ) as an analog of the universal lower bound for E h ( n, M, − from [9]. We will usethe notation S j = j X i =0 r i = j X i =0 ( q − i (cid:18) ni (cid:19) for j ∈ { k − , k, k + 1 } . Theorem 5.2
Let n , q , k , ℓ , and s satisfy the conditions (18) , (20) , (24) , and (26) and supposethe ( k, ℓ ) -strengthened Krein condition holds. Then the polynomial f n,ℓ,s k ( t ) belongs to F n,ℓ,s and,therefore, A q ( n, ℓ, s ) ≤ f n,ℓ,s k (1) f = 1 ρ k +1 = L k ( n, ℓ, s ) , (34) where L k ( n, ℓ, s ) := S k (cid:16) Q ,ℓk − ( s ) − Q ,ℓk ( s ) (cid:17) r k +1 Q k +1 ( ℓ ) Q ,ℓk − ( s ) S k +1 ( Q , k +1 ( ℓ ) − Q , k ( ℓ ) ) − r k Q k ( ℓ ) Q ,ℓk ( s ) S k − ( Q , k ( ℓ ) − Q , k − ( ℓ ) ) . Furthermore, for fixed ℓ , for h being an absolutely monotone function on [ − , , and for M determined by f n,ℓ,s k (1) = M f , the Hermite interpolant g n,ℓ,M k ( t ) := H (( t − s ) f n,ℓ,s k ( t ); h ) belongs to G ( h ) n,ℓ , and, therefore, E h ( n, M, ℓ ) ≥ M ( M g − g n,ℓ,M k (1))= M k X i =0 ρ i h ( α i ) . (35) The notation g = H ( f ; h ) signifies that g is the Hermite interpolant to the function h at the zeros (taken withtheir multiplicity) of f . he bounds (34) and (35) can be attained only simultaneously by codes which have all theirinner products in the roots of f n,ℓ,s k ( t ) and which are, in addition, k -designs in F nq .Proof. It follows from the definitions (25) and (28) that the polynomial f n,ℓ,s k ( t ) can be writtenas c ( t − ℓ ) (cid:16) Q ,ℓk ( t ) + c Q ,ℓk − ( t ) (cid:17) k − X i =0 r ,ℓi Q ,ℓi ( t ) Q ,ℓi ( s ) , where r ,ℓi > , i = 0 , , . . . , k − , and the constants c = (1 − s ) / (1 + c ) R ,ℓk − (1 , s ) and c = − Q ,ℓk ( s ) /Q ,ℓk − ( s ) are positive under the assumptions for ℓ and s . Since Q ,ℓi ( s ) > for ≤ i ≤ k − , the polynomial f n,ℓ,s k ( t ) becomes a positive linear combination of polynomials ( t − ℓ ) Q ,ℓi ( t ) Q ,ℓj ( t ) , where i ∈ { k, k − } and j ≤ k − . Therefore f n,ℓ,s k ( t ) ∈ F > because of the ( k, ℓ ) -strengthened Krein condition. This and the obvious f n,ℓ,s k ( t ) ≤ for every t ∈ [ ℓ, s ] impliesthat f n,ℓ,s k ( t ) ∈ F n,ℓ,s .To compute the ratio f n,ℓ,s k (1) /f we write f as in (31) and then use the representation of ( t − ℓ ) Q ,ℓj ( t ) by the Christoffel-Darboux formula (see (19) and (21)) for j = k − and k . Theintegrand becomes a linear combination of the polynomials Q , i ( t ) , i = k − , k, k + 1 . Since Z − Q , j ( t ) dµ n ( t ) = Z − T j ( t, T j (1 , dµ n ( t ) = 1 S j , after simplifications we obtain the explicit form of the bound (34).We proceed with the energy bound. Denote by t ≤ t ≤ · · · ≤ t k the zeros of f n,ℓ,s k ( t ) in increasing order and counting their multiplicity; i.e., t := α = ℓ , t i = t i +1 := α i , i =1 , . . . , k − , and t k := α k = s . Then the Newton interpolation formula gives that the polynomial g n,ℓ,M k ( t ) is a linear combination with nonnegative coefficients of the constant 1 and the partialproducts m Y j =1 ( t − t j ) , m = 1 , , . . . , k. Since t i , i = 1 , . . . , k , are the roots of Q ,ℓk ( t )+ αQ ,ℓk − ( t ) (see (25)) it follows from [14, Theorem3.1] that the partial products Q mj =1 ( t − t j ) , m = 1 , . . . , k − , have positive coefficients whenexpanded in terms of the polynomials Q ,ℓi ( t ) , i = 0 , , . . . , k − . Therefore g n,ℓ,M k ( t ) is a linearcombination with positive coefficients of terms ( t − ℓ ) Q ,ℓi ( t ) Q ,ℓj ( t ) , i, j ∈ { , , . . . , k − } , andthe last partial product which is in fact f n,ℓ,s k ( t ) . Now g n,ℓ,M k ( t ) ∈ F > follows from the validity ofthe ( k, ℓ ) -strengthened Krein condition and from f n,ℓ,s k ( t ) ∈ F > , obtained in the first part of theproof.Multiple application of the Rolle’s theorem implies that g n,ℓ,M k ( t ) ≤ h ( t ) for every t ∈ [ ℓ, andtherefore g n,ℓ,M k ( t ) ∈ G ( h ) n,ℓ . The explicit form of the bound (35) via the weights ρ i and the nodes Also known as orthogonal arrays of strength k . i follows from the quadrature formula (30) applied for g n,ℓ,M k ( t ) and the interpolation conditions g n,ℓ,M k ( α i ) = h ( α i ) , i = 0 , , . . . , k .There are two kinds of conditions for attaining the general linear programming bounds (2)and (3) (see, for example, [25, Eqs. (32)-(33)] for (2)). First, the inner products of distinct pointsof any attaining code must be among the zeros of the polynomial f ( t ) in (2) or the abscissasof the touching/intersection points of the polynomial g ( t ) and the potential function h ( t ) in (3).Second, the complementary slackness conditions f i B ′ i = 0 (or g i B ′ i = 0 ) for i = 1 , , . . . , n , where ( B ′ , B ′ , . . . , B ′ n ) is the MacWilliams transform of the attaining code, have to be satisfied.By our construction, the roots of the polynomial f n,ℓ,s k ( t ) coincide exactly with the abscissasof the touching/intersection points of the graphs of g n,ℓ,M k ( t ) and h ( t ) . Further, f n,ℓ,s k ( t ) ∈ F > implies (and g n,ℓ,M k ( t ) ∈ F > does as well) that B ′ i = 0 for i = 1 , , . . . , k ; i.e., any attaining codehas to be a k -design.Therefore, the bounds (34) and (35) can be attained only simultaneously by codes whichhave all their inner products in the roots of f n,ℓ,s k ( t ) (equivalently, in the abscissas of the touch-ing/intersection points of the graphs of g n,ℓ,M k ( t ) and h ( t ) ) and which are, in addition, k -designsin F nq . This completes the proof. (cid:3) The bound (34) was obtained and investigated for k = 1 (in our notations) and the corre-sponding ℓ and s by Helleseth, Kløve and Levenshtein [21]. In that paper, comparisons with theLevenshtein bound (see [25]) obtained by polynomials of degrees 2 and 3, and detailed descrip-tions of all known codes attaining L ( n, ℓ, s ) can be found. We discuss some examples from [21]in Section 10. The bound (35) for k = 1 is given by E h ( n, M, ℓ ) ≥ M ( ρ h ( ℓ ) + ρ h ( s ) , where ρ and ρ can be computed as shown in Example 6.3 below.For k > , it does not seem customary to consider the bounds (34) and (35) for fixed k andvarying ℓ and s . Instead, in Sections 7 and 10.1 we describe them as a system of bounds for fixed ℓ > − and varying k = 1 , , . . . and corresponding s ∈ I ( ℓ ) k ⊂ ( t ,ℓk − ,k − , , like the Levenshteinbound is described with fixed ℓ = − .The optimality of the bounds (34) and (35) will be discussed in Section 7. Remark 5.3
The above proof of the bound (35) does not require M to be integer. In particular,the expression at the right hand side of (35) is defined for any real M ∈ [2 , q n ] . This is customaryin certain investigations. Like in the case ℓ = − (see Theorem 5.55 and Remark 5.58 in [26]; also [6] for details), codeswhich attain the bounds from Theorem 5.2 have special combinatorial and geometric properties.Also, it is important that the bounds (34) and (35) can be attained only simultaneously since theconditions of their attaining coincide.The conditions α i ∈ T n , i = 1 , , . . . , k − , are quite restrictive. For example, they say thatall roots of Q ,ℓ,sk − ( t ) belong to T n . In particular, the roots of Q ,ℓ,sk − ( t ) must be all rational which17s usually a good starting point for deep algebraic investigation. However, in this section we focuson the combinatorial meaning of the fact that all inner products of attaining codes must belongto the set { α , α , . . . , α k } . Definition 6.1
Let C ⊂ F nq be a code. For fixed x ∈ C and t n − i ∈ T n , i ∈ { , , . . . , n } , denoteby A i ( x ) := |{ y ∈ C : h x, y i = t n − i }| , the number of the points of C at distance i from x . The system of nonnegative integers ( A i ( x ) : i = 0 , , . . . , n ) is called distance distribution of C with respect to x . It is clear that A ( x ) = 1 and that A i ( x ) = 0 is possible only for i ∈ { d, d − , . . . , D − , D } ,where d and D are the minimum and maximum distance of C , respectively (recall that s = 1 − d/n and ℓ = 1 − D/n ). We show that for codes attaining (34) and (35) the whole distance distributioncan be computed.When dealing with distance distributions, it is convenient to use the following characteristicproperty of designs in polynomial metric spaces (see [25] for Hamming spaces; Equation (1.10)in [20] for the general case of polynomial metric spaces). A code C ⊂ F nq is a τ -design if and onlyif X y ∈ C f ( h x, y i ) = f | C | (36)holds for every x ∈ F nq and every real polynomial f ( t ) of degree at most τ . Theorem 6.2
If a code C ⊂ C n,q ( ℓ, s ) attains the bounds (34) and (35) , then its distance distri-bution with respect to any point x ∈ C does not depend on the choice of x and can be computedfrom a system of linear equations. Explicitly, we have A α i = A α i ( x ) = ρ i | C | (cid:18) = ρ i ρ k +1 (cid:19) , i = 0 , , . . . , k. Proof.
Let C be a code that attains the bounds (34) and (35). By Theorem 5.2 the code C isa k -design, so (36) holds. Fixing x ∈ C and grouping the terms with the same t n − i in the lefthand side we write (36) as D X i = d A i ( x ) f ( t n − i ) = f | C | . (37)In our case A i ( x ) = 0 is possible only if t n − i ∈ { α = ℓ, α , . . . , α k − , s = α k } (see Theorem5.2). Setting consecutively f ( t ) = 1 , t, t , . . . , t k in (37) yields the Vandermonde-type system k X i =0 A α i ( x ) α ui = b u | C | , u = 0 , , . . . , k, (38)where b u = Z − t u dµ n
18s the zeroth coefficient in the Krawtchouk expansion of t u , u = 0 , , . . . , k .Since the solution of (38) is unique, it follows that the distance distribution { A α i ( x ) : i =0 , , . . . , k } does not depend on the choice of x ∈ C and can be computed from the system (38)(so it is uniquely determined by the parameters n , q , ℓ , s , and | C | = L k ( n, ℓ, s ) ). Thus we canomit x in the notation of the distance distributions of C .The combination of (38) and the quadrature formula (30) gives explicit formulas for thedistance distributions. Indeed, setting (again!) the polynomials f ( t ) = 1 , t, t , . . . , t k in (30)produces the system ρ k +1 + k X i =0 ρ i α ui = b u , u = 0 , , . . . , k. (39)Multiplying all equations of (39) by | C | and taking into account that ρ k +1 | C | = ρ k +1 L k ( n, ℓ, s ) = 1 by (34), we obtain the system (38) again but with unknowns ρ i | C | . The solutions of both systemsmust coincide; i.e., A α i = ρ i | C | , i = 0 , , . . . , k , as required. (cid:3) Of course, the formulas from Theorem 6.2 have to produce nonnegative integers. Thus theyin fact yield strong necessary conditions for existence of codes attaining (34) and (35).The above approach works (to some extent) also for the external (when x ∈ F nq \ C ) distancedistributions of C with respect to x . Then it yields a system of k + 1 equations with respectto n unknowns A ( x ) , A ( x ) , . . . , A n ( x ) (note that A ( x ) = 0 ). Typically, n is quite larger than k + 1 and our system has many solutions. However, the solutions belong to a finite set and itis possible to find them for subsequent analysis (see, for example, [11, 12]). Such computationscould yield upper bounds on the covering radius of codes attaining (34) and (35). In fact, thisapproach works in general for designs in F nq as well (see [11, 12]).We remark also that the computations of distance distributions of attaining codes allows easyderivation of the energy of these codes. Example 6.3
We show how Theorem 6.2 works for k = 1 . Assume that C ⊂ C n,q ( ℓ, s ) attainsthe bound L ( n, ℓ, s ) . Then the system (38) (for k = 1 ) is solved explicitly as follows. We have b = (2 − q ) /q and (38) becomes (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A ℓ + A s = | C | − ℓA ℓ + sA s = (2 − q ) | C | q − , whence we obtain A ℓ = q (1 + s )( | C | − − | C | q ( s − ℓ ) ,A s = 2 | C | − q (1 + ℓ )( | C | − q ( s − ℓ ) . Now ρ = A ℓ / | C | and ρ s = A s / | C | are computed in turn and the energy of C (attaining the bound (35) ) is given by E h ( C ) = | C | ( ρ h ( ℓ ) + ρ h ( s ))= | C | ( A ℓ h ( ℓ ) + A s h ( s )) . On the optimality of the bounds (34) and (35)
In this section we assume that n , q , k , and ℓ ∈ (cid:16) t , k +1 , , t , k, (cid:17) are fixed and there exists an interval I ( ℓ ) k ⊂ ( t , k,k , such that for every s ∈ I ( ℓ ) k the bounds (34) and (35) are optimal in the followingsense L k ( n, ℓ, s ) = min { L j ( n, ℓ, s ) : j ≥ , s ∈ I ( ℓ ) k } ; i.e., the bound L k ( n, ℓ, s ) is optimal for every s ∈ I ( ℓ ) k among all bounds L j ( n, ℓ, s ) (if any with j = k ) . Then the image of I ( ℓ ) k under the function L k ( n, ℓ, s ) (which is continuous in s ) is asubinterval of ( L k ( n, ℓ, t , k,k ) , q n ) denoted by J ( ℓ )2 k .For s ∈ I ( ℓ ) k and positive integer j , we define R n,ℓj ( s ) := 1 L k ( n, ℓ, s ) + k X i =0 ρ i Q j ( α i ) . Similarly, for M ∈ J ( ℓ )2 k and positive integer j , we define S n,ℓj ( M ) := 1 M + k X i =0 ρ i Q j ( α i ) , where the parameters ( ρ i , α i ) ki =0 come from fixing s ∈ I ( ℓ ) k by the equality M = L k ( n, ℓ, s ) . Notethat the values of S n,ℓj ( M ) for integers M ∈ J ( ℓ )2 k are just particular values of R n,ℓj ( s ) for s ∈ I ( ℓ ) k .On the other hand, it is clear that M could be considered as real variable whenever this facilitatesan analysis. Remark 7.1
The quadrature formula (30) applied for Q ( n,q ) j ( t ) with ≤ j ≤ k implies immedi-ately that R n,ℓj ( s ) = 0 for all s ∈ I ( ℓ ) k and that S n,ℓj ( M ) = 0 for all M ∈ J ( ℓ )2 k . Thus we sometimesassume (to avoid trivialities) in what follows that j ≥ k + 1 . The next theorem gives necessary and sufficient conditions for existence of better bounds than(34) and (35) (obtained by polynomials from F n,ℓ,s and G ( h ) n,ℓ , respectively). Its part (a) is acounterpart of Theorem 5.47 from [26] (see also [7, Theorem 3.1]) and its part (b) is a counterpartof Theorem 5.1 from [9]. Theorem 7.2 (a)
Given n , q , ℓ , k , and s ∈ I ( ℓ ) k , the bound (34) can be improved by a polynomialfrom F n,ℓ,s if and only if there exists a positive integer j ≥ k + 1 such that R n,ℓj ( s ) < . Inparticular, if R n,ℓj ( s ) ≥ for every j ≤ m then (34) cannot be improved by a polynomial from F n,ℓ,s of degree at most m . (b) Given n , q , ℓ , k , M ∈ J ( ℓ )2 k , and a strictly absolutely monotone h , the bound (35) can beimproved by a polynomial from G ( h ) n,ℓ if and only if there exists a positive integer j ≥ k + 1 suchthat S n,ℓj ( M ) < . In particular, if S n,ℓj ( M ) ≥ for every j ≤ m then (35) cannot be improved bya polynomial from G ( h ) n,ℓ of degree at most m . roof. (a) Assume that R n,ℓj ( s ) ≥ for every positive integer j . Let f ( t ) ∈ F n,ℓ,s and f ( t ) = u ( t ) + X j ≥ k +1 f j Q j ( t ) , where u ( t ) has degree at most k and zeroth coefficient u . Note that f ( α i ) ≤ for i = 0 , , . . . , k , f j ≥ for j ≥ k + 1 , and f = u . Applying (30) to u ( t ) and using that L k ( n, ℓ, s ) = 1 /ρ k +1 ,we obtain f = u = u (1) L k ( n, ℓ, s ) + k X i =0 ρ i u ( α i )= f (1) L k ( n, ℓ, s ) + k X i =0 ρ i f ( α i ) − X j ≥ k +1 f j R n,ℓj ( s ) ≤ f (1) L k ( n, ℓ, s ) . Therefore, f (1) /f ≥ L k ( n, ℓ, s ) ; i.e., f ( t ) does not produce better bound than (34).Let, conversely, R n,ℓj ( s ) < for some j ≥ k +1 . We construct a degree j improving polynomialof the form v ( t ) = ( a ( t ) + c ) f n,ℓ,s k ( t ) = j X i =0 v i Q i ( t ) , where the number c and the polynomial a ( t ) of degree j − k will be properly chosen.The polynomial a ( t ) is immediate – applying polynomial division we consider the uniquepolynomials a ( t ) (quotient) and b ( t ) (remainder) such that Q j ( t ) = a ( t ) f n,ℓ,s k ( t ) + b ( t ) , where the remainder b ( t ) has degree at most k − . Let b ( t ) = k − X i =0 b i Q i ( t ) , f n,ℓ,s k ( t ) = k X i =0 f i Q i ( t ) be the Krawtchouk expansions of b ( t ) and f n,ℓ,s k ( t ) , respectively. Then it is easy to see that c := max (cid:26) − min t ∈ [ ℓ,s ] a ( t ) , max ≤ i ≤ k − b i f i , (cid:27) (recall that f i > for every i = 0 , , . . . , k ) guaranties that v ( t ) ∈ F n,ℓ,s .Since the polynomial v ( t ) − Q j ( t ) = cf n,ℓ,s k ( t ) − b ( t ) k , its zeroth coefficient v (which coincides with the zeroth coefficient of v ( t ) )can be computed from (30). We have consecutively v = cf n,ℓ,s k (1) − b (1) L k ( n, ℓ, s ) + k X i =0 ρ i (cid:16) cf n,ℓ,s k ( α i ) − b ( α i ) (cid:17) = v (1) − L k ( n, ℓ, s ) − k X i =0 ρ i Q j ( α i )= v (1) L k ( n, ℓ, s ) − R n,ℓj ( s ) > v (1) L k ( n, ℓ, s ) . Therefore v (1) /v < L k ( n, ℓ, s ) ; i.e., v ( t ) improves on L k ( n, ℓ, s ) , which completes the proof ofthe sufficiency.(b) Suppose that S n,ℓj ( M ) ≥ for every positive integer j . Any polynomial g ( t ) ∈ G ( h ) n,ℓ can bewritten as g ( t ) = u ( t ) + X j ≥ k +1 g j Q j ( t ) for some polynomial u ( t ) of degree at most k with zeroth coefficient u . We have g ( α i ) ≤ h ( α i ) for i = 0 , , . . . , k , g j ≥ for every j ≥ k + 1 , and g = u . Therefore, using (30) for u ( t ) (recallthat M = L k ( n, ℓ, s ) ), we consecutively obtain M g − g (1) = M u − g (1)= M k X i =0 ρ i u ( α i ) − X j ≥ k +1 g j = M k X i =0 ρ i g ( α i ) − X j ≥ k +1 g j Q j ( α i ) − X j ≥ k +1 g j = M k X i =0 ρ i g ( α i ) − M X j ≥ k +1 g j M + k X i =0 ρ i Q j ( α i ) ! = M k X i =0 ρ i g ( α i ) − M X j ≥ k +1 g j S n,ℓj ( M ) ≤ M k X i =0 ρ i h ( α i ) , where, for the last inequality, we used S n,ℓj ( M ) ≥ for j ≥ k + 1 . Hence the bound, producedby g ( t ) , does not improve on (35).Conversely, assume that h is strictly absolutely monotone and suppose that S n,ℓj ( M ) < forsome positive integer j ≥ k + 1 . We are going to improve (35) by using a polynomial v ( t ) = εQ j ( t ) + a ( t ) = j X i =0 v i Q i ( t ) , ε > and the polynomial a ( t ) of degree at most k will be properly chosen.Denote ˜ h ( t ) := h ( t ) − εQ j ( t ) and select ε such that ˜ h ( i ) ( t ) ≥ on [ ℓ, for all i = 0 , , . . . , j . This choice of ε is possible sincethe function h is strictly absolutely monotone. Since ˜ h ( i ) ( t ) = h ( i ) ( t ) > for i > j the function ˜ h ( t ) is absolutely monotone.Now the polynomial a ( t ) is chosen to be the Hermite interpolant of the new function ˜ h atthe nodes ℓ = α (simply) and α i , i = 1 , , . . . , k , (doubly) exactly as the original g n,ℓ,M k ( t ) does.Then we can infer as in Theorem 5.2 that a ( t ) ∈ G (˜ h ) n,ℓ implying that v ( t ) ∈ G ( h ) n,ℓ .It remains to prove that v ( t ) gives a bound which is better than (35) indeed. Let a ( t ) = k − X i =0 a i Q i ( t ) and note that v = a and v (1) = a (1) + ε . We multiply by ρ i and sum up the interpolationequalities for a ( t ) to compute k X i =0 ρ i a ( α i ) = k X i =0 ρ i h ( α i ) − ε k X i =0 ρ i Q j ( α i ) . Since M k X i =0 ρ i a ( α i ) = M a − a (1) by (30) and M k X i =0 ρ i Q j ( α i ) = M S n,ℓj ( M ) − by the definition of the function S n,ℓj ( M ) , we obtain M a − a (1) = M k X i =0 ρ i h ( α i ) + ε − εM S n,ℓj ( M ) which yields M v − v (1) = M k X i =0 ρ i h ( α i ) − εM S n,ℓj ( M ) > M k X i =0 ρ i h ( α i ) . The last inequality means that the polynomial v ( t ) gives better than (35) bound. (cid:3) Remark 7.1 and Theorem 7.2 give the following optimality property of the bounds (34) and(35). 23 orollary 7.3
None of the bounds (34) and (35) can be improved by using polynomials from F n,ℓ,s and G ( h ) n,ℓ , respectively, of degree at most k . The corresponding optimality results for the case ℓ = − were proved for the maximum codeproblem by Sidelnikov [27] (see also [24]) and for the minimum energy problem by the authors [9].We provide another formula for the test functions. We use the notations Q j ( t ) := j X i =0 a j,i t i for the coefficients of the Krawtchouk polynomials, S u = 1 L k ( n, ℓ, s ) + k X i =0 ρ i α ui , and recall that b u := R − t u dµ n as in the proof of Theorem 6.2. Lemma 7.4
With the above notations, j X i =0 a j,i b i = 0 for very positive integer j .Proof. This is the zeroth coefficient of Q j ( t ) which is, of course, equal to 0. (cid:3) Lemma 7.5
With the above notations, b u = S u for every u = 0 , , . . . , k .Proof. This follows from the quadrature formula (30) applied with the polynomial t u . (cid:3) Theorem 7.6
For every s ∈ I ( ℓ ) k and positive integer j > k , R n,ℓj ( s ) = j X u =2 k +1 a j,u ( S u − b u ) and, correspondingly, S n,ℓj ( M ) = j X u =2 k +1 a j,u ( S u − b u ) with parameters coming from M = L k ( n, ℓ, s ) ∈ J ( ℓ )2 k as in Theorem 5.2. roof. It is enough to prove the formula for R n,ℓj ( s ) . Grouping the powers of α i in the definitionof R n,ℓj ( s ) and using Lemma 7.4 yield R n,ℓj ( s ) = j X u =0 a j,u ( S u − b u ) . Now Lemma 7.5 implies the required identity. (cid:3)
For fixed n , q , ℓ and k , there are only finitely many s ∈ T n ∩ I ( ℓk and finitely many M ∈ J ( ℓ ) k .Thus a numerical investigation of the signs of the functions R n,ℓj ( s ) and S n,ℓj ( M ) can be performed.We conclude this section with a few comments on the possibility for using higher degreepolynomials.Corollary 7.3 implies (like in the case ℓ = − ) that improvements of the bounds (34) and (35)by polynomials are only possible for degrees higher than k . We refer to (34) and (35) to as firstlevel bounds and call second level bounds any improvement by polynomials from F n,ℓ,s or G ( h ) n,ℓ .In the proof of Theorem 7.2 we, in fact, produced improving polynomials. However, thenumerical experiments show that these are marginal and are never optimal like the first levels are.A detailed second level universal bounds based on Levenshtein-type quadratures that generalize(30) will be developed in a future work (see [10] for the spherical codes case when ℓ = − ). Corollary 7.3 shows that the bounds (34) and (35) cannot be improved by using polynomials from F n,ℓ,s and G ( h ) n,ℓ , respectively, of degree at most k . However, the requirements f ( t ) ≤ (or g ( t ) ≤ h ( t ) , respectively) for every t ∈ [ ℓ, s ] (for every t ∈ [ ℓ, , respectively) are stronger than reallynecessary. What we need in fact, is f ( t ) ≤ (or g ( t ) ≤ h ( t ) , respectively) for every t ∈ [ ℓ, s ] ∩ T n (for every t ∈ [ ℓ, ∩ T n , respectively). Of course, we always have { ℓ, s } = { t n − D , t n − d } ⊂ T n ,but the roots α , α , . . . , α k − of the polynomial f n,ℓ,s k ( t ) are not necessarily in the set T n . Thismakes a difference for k > allowing a natural relaxation of our linear programming problems.We describe a modification of the polynomials f n,ℓ,s k ( t ) and g n,M,ℓ k ( t ) which is going to producebetter bounds provided the new polynomials are still good for linear programming.We replace the double roots α , α , . . . , α k − (the touching points, respectively) with theirclosest neighbours from T n . More precisely, if α i ∈ ( t j − , t j ) for some integer j ∈ { n − D, n − D +1 , . . . , n − d } , then we replace the double zero α i of f n,ℓ,s k ( t ) by two simple zeros γ i − = t j − and γ i = t j . If α i = t j , then one can try both ( γ i − , γ i ) = ( t j − , t j ) and ( t j , t j +1 ) . Finally, setting γ := ℓ and γ k − := s , we define our refining polynomial for the maximum code problem to be f ref ( t ) := k − Y i =0 ( t − γ i ) = k X i =0 f i Q i ( t ) . Then g ref ( t ) := H ( f ref ( t )( t − t n − d − ); h ) = k X i =0 g i Q i ( t ) g ref we may haveintersections at t n − d and at t n − d − or t n − d +1 instead of touching at s = t n − d .The above construction obviously preserves the conditions for f ref ( t ) and g ref ( t ) for stayingfeasible at the points of T n ; i.e., we still have f ref ( t ) ≤ , for every t ∈ T n ∩ [ ℓ, s ] and g ref ( t ) ≤ h ( t ) for every t ∈ T n ∩ [ ℓ, ).Therefore, only the positive definiteness of f ref ( t ) and g ref ( t ) remains to be investigated. Weremark that the new polynomials have obviously f k > and g k > . Moreover, it followsfrom the construction that for every i = 0 , , . . . , k we have f ref ( α i ) ≥ and g ref ( α i ) ≥ h ( α i ) with equality if and only if α i ∈ T n . Thus the quadrature formula (30) implies that the newpolynomials have f > and g > . Furthermore, it also implies that f n,ℓ,s k (1) f ≥ f ref (1)( f ref ) , g − g n,ℓ,M k (1) M ≤ ( g ref ) − g ref (1) M , so the bounds (34) and (35) are indeed improved as claimed above (provided the new polynomialsare still feasible).Numerical investigation of the remaining feasibility conditions f i ≥ ( g i ≥ ) for i =1 , , . . . , k − show that they are satisfied in numerous cases. Moreover, numerics lead us to thefollowing conjecture concerning the relaxation of the linear programing over the discrete subset [ ℓ, s ] ∩ T n or [ ℓ, ∩ T n as introduced above. Conjecture 8.1
For fixed q ≥ , n , and ℓ there exists a constant s ( q, n, ℓ ) such that whenever s ∈ [ − , s ( q, n, ℓ )) ∩ T n (that is large enough d/n = (1 − s ) / ) the new polynomials f ref and g ref solve the relaxed linear programming in the context above In other words, we conjecture that for most parameters the roots α , α , . . . , α k − of theLevenshtein-type polynomial f n,ℓ,s k ( t ) are the best approximation of the optimal nodes for generallinear programming. This implies significantly faster computation compared, for example, tothe simplex method (see [5, 30]). More detailed investigation in this direction will be consideredelsewhere. For given q , n , M , s , and ℓ , denote by C n,q ( M, ℓ, s ) := { C ⊂ F nq : | C | = M, s ( C ) = s, ℓ ( C ) = ℓ } F nq of cardinality M ∈ [2 , q n ] , minimum distance d = n (1 − s ) / and diameter D = n (1 − ℓ ) / . In this section we derive a universal upper bound on the quantity U h ( n, M, ℓ, s ) := max { E h ( C ) : C ∈ C n,q ( M, ℓ, s ) } , where h is absolutely monotone.The linear programming problem in this case can be formulated as follows U h ( n, M, ℓ, s ) ≤ min p ( t ) ∈Q ( h ) n,M,ℓ,s M ( p M − p (1)) , (40)with p ( t ) = n X i =0 p i Q i ( t ) , Q ( h ) n,M,ℓ,s := { p ( t ) ∈ F ≤ : p ( t ) ≥ h ( t ) , t ∈ [ ℓ, s ] } , where F ≤ := { p ( t ) : p i ≤ , i = 1 , , . . . , n } .We construct polynomials which belong to the set Q ( h ) n,M,ℓ,s and therefore provide upper boundsfor U h ( n, M, ℓ, s ) by (40). Let s ∈ I ( ℓ ) k , α = ℓ, α , . . . , α k − , α k = s be the roots of f n,ℓ,s k ( t ) asabove, and ρ , ρ , . . . , ρ k are the corresponding weights from (30). Note that the parameters ofthe quadrature (30) no longer come with M but with s instead.We consider p n,M,ℓ,s k ( t ) := − λf n,ℓ,s k ( t ) + g L ( t ) = k X i =0 p i Q i ( t ) , (41)where λ > is a parameter (to be determined and optimized later) and g L ( t ) := H ( f n,ℓ,s k ( t ); h ( t )) is the Hermite interpolation polynomial to the function h ( t ) that agrees with h ( t ) exactly in theroots of the Levenshtein-type polynomial f n,ℓ,s k ( t ) (counted with their multiplicities).Note that deg( g L ) ≤ k − and therefore deg( p n,M,ℓ,s k ) = 2 k . Let f n,ℓ,s k ( t ) = k X i =0 f i Q i ( t ) , g L ( t ) = k − X i =0 g i Q i ( t ) be the Krawtchouk expansions of f n,ℓ,s k ( t ) and g L ( t ) , respectively.The next theorem is the main result in this section. Theorem 9.1
Let n , q , k , ℓ , and s be such that the conditions of Theorem 5.2 are fulfilled andlet C ∈ C n,q ( M, ℓ, s ) . Then E h ( C ) ≤ p n,M,ℓ,s k (1) M ( M − L k ( n, ℓ, s )) L k ( n, ℓ, s ) + M k X i =0 ρ i h ( α i ) or every large enough λ . In particular, U h ( n, M, ℓ, s ) ≤ U k ( n, M, ℓ, s ) (42) = p n,M,ℓ,s k (1) M ( M − L k ( n, ℓ, s )) L k ( n, ℓ, s ) + M k X i =0 ρ i h ( α i ) , where λ is chosen by λ := max (cid:26) g i f i : 1 ≤ i ≤ k − (cid:27) . (43) The bound (42) can be attained only by codes which have all their inner products in the rootsof f n,ℓ,s k ( t ) and p i B ′ i = 0 for i = 1 , , . . . , k , ( B ′ , B ′ , . . . , B ′ n ) is the MacWilliams transform ofthe attaining code.Proof. Since p i = − λf i + g i and f i > for every i = 0 , , . . . , k , it follows that large enough λ > will make p i ≤ for every ≤ i ≤ k − . Adding the obvious p k < , we conclude that p n,M,ℓ,s k ( t ) ∈ F ≤ .Moreover, the absolute monotonicity of h ( t ) and the interpolation conditions for g L ( t ) implythat g L ( t ) ≥ h ( t ) for t ∈ [ ℓ, s ] . Since f n,ℓ,s k ( t ) ≤ for t ∈ [ ℓ, s ] and p n,M,ℓ,s k ( α i ) = g L ( α i ) = h ( α i ) for every i = 0 , , . . . , k , it follows from (41) that p ( t ) ≥ h ( t ) for every t ∈ [ ℓ, s ] (whatever λ > is).Therefore p n,M,ℓ,s k ( t ) ∈ Q ( h ) n,M,ℓ,s for large enough λ and it remains to compute the correspondingbound.We first note that L k ( n, ℓ, s ) ≥ M follows from the monotonicity of the bound (34). Express-ing p by the quadrature formula (30) and using the interpolation conditions we obtain p M − p n,M,ℓ,s k (1)= (cid:18) ML m ( n, s ) − (cid:19) p n,M,ℓ,s k (1) + M k X i =0 ρ i h ( α i ) , whence we get (42) with λ still to be optimized. The dependence of the right hand side of (42) onthe parameter λ comes from p n,M,ℓ,s k (1) only. Since p n,M,ℓ,s k (1) is linear and increasing with respectto λ , the best bound is obtained when λ is chosen as in (43); i.e., when it is the smallest possiblereal number which satisfies all conditions p i = − λf i + g i ≤ , i = 1 , , . . . , k − , simultaneously.Note that λ > by (43) since at least one of the ratios, g k − /f k − , is positive.The description of necessary conditions for attaining codes is similar to that in Theorem 5.2. (cid:3) Corollary 9.2
The energy of every code from C n,q ( M, ℓ, s ) belongs to the interval [ L, U ] , where L := M k X i =0 ρ ′ i h ( α ′ i ) (the parameters are determined by M = L k ( n, ℓ, s ′ ) ; i.e., by M ) and U := M ( M − L k ( n, ℓ, s ) f (1)) L k ( n, ℓ, s ) + M k X i =0 ρ i h ( α i ) the parameters are determined by f n,ℓ,s k ( t ) ; i.e., by s ), respectively. A modification that adds ℓ to the interpolation nodes for g L ( t ) (so ℓ becomes a double node)works in a similar way as in Theorem 9.1. However, it is not difficult to prove that the boundproduced is the same.It is clear that the refining technique from Section 8 can be applied for improving the bounds(42) and, consequently, for shrinking the interval [ L, U ] from Corollary 9.2.
10 Examples q = 2 and ℓ = − /n We show as a typical example the mix of the Levenshtein bounds (see [26, Table 6.3]) and ourLevenshtein-type bounds (34) for A ( n, ℓ, s ) , where ℓ = t = − n = 2 − nn is fixed (this ℓ corresponds to D = n − , the second largest possible diameter). The fist fourbounds (two Levenshtein bounds and our bounds for k = 1 and k = 2 ) are explicitely stated.For s ∈ [ ℓ, − /n ] the first Levenshtein bound A ( n, − nn , s ) ≤ s − s is valid. Our bound (34) for k = 1 A ( n, − nn , s ) ≤ L ( n, − nn , s ) = 2(1 − s )( n − − ( n − s is valid for s ∈ (cid:16) t , , , t ,ℓ , i = (cid:18) − n , n ( n − (cid:21) . Then the next Levenshtein bound A ( n, − nn , s ) ≤ (1 − s ) n (2 + ( n + 1) s )1 − ns comes for s ∈ (cid:16) t , , , t , , i = (cid:18) , √ n − − n (cid:21) . Our bound (34) for k = 2 is given by A ( n, − nn , s ) ≤ L ( n, − nn , s )= 2(1 − s ) n ( n − s ( n −
4) + n − nAs + 9 Bs + Cs + D , A = n − n + 25 n − ,B = 29 n − n + 351 n − ,C = n − n + 369 n − n − n + 972 ,D = − n + 36 n − n + 81 n + 162 . It is valid for s ∈ (cid:16) t , , , t ,ℓ , i = √ n − − n , √ n − n + 41 n − n ( n − . The even weight codes C ⊂ F nq , where n = 2 m + 1 is odd, attain the bound L m − (cid:18) m + 1 , − nn , n − n (cid:19) = 2 m and the corresponding bound (35). k = 1 and k = 2 As mentioned above, the case k = 1 was considered by Helleseth, Klove and Levenshtein [21]. Inour notations, their bound (see [21, Theorem 1]) is A q ( n, ℓ, s ) ≤ LL + 4( q − − n ) + 2 nq ( q − s + ℓ ) , where L = nq (1 − s )(1 − ℓ ) .Examples with ℓ = − are covered by the Levenshtein bounds (see, for example, Table 1in [25]). For ℓ > − and k = 1 , we extract the following examples from [21].For n = 6 , q = 2 , ℓ = − / ( D = 4 ), s = 1 / ( d = 2 ) an explicit nonlinear code in[21, Example 1] has cardinality M = 16 = L (6 , − / , / . For n = 5 , q = 2 , ℓ = − / ( D = 4 ), s = 1 / ( d = 2 ) the binary [5 , , even weight code in [21, Example 2] has cardinality M = 16 = L (5 , − / , / . For n = 56 , q = 3 , ℓ = − / ( D = 45 ), s = − / ( d = 36 ) theHill (ternary) projective cap (see [22]) has cardinality M = 729 = L (56 , − / , − / and for n = 78 , q = 4 , ℓ = − / ( D = 64 ), s = − / ( d = 56 ) the Hill (quaternary) projectivecap (see [23]) has cardinality M = 729 = L (56 , − / , − / . All these codes have h -energieswhich attain (35) for the corresponding parameters and for every absolutely monotone h . In allcases the distance distributions of the attaining codes are easily computed by Theorem 6.2 (seeExample 6.3).The ovoids in PG (3 , q ) (see [13]) are codes C ⊂ F nq with parameters n = q + 1 , d = q − q, D = q , | C | = q . s = 1 + 2 q − q q , ℓ = 1 − q q and the bound (34) is attained, | C | = M = q = L (cid:18) q + 1 , − q q , q − q q (cid:19) . The distance distribution of these codes can be computed by the system (38) (for k = 1 ; as inExample 6.3)) and is given by A ℓ = ( q − q + 1) = n ( q − ,A s = q ( q − q + 1) = nd. Thus, in turn we find ρ = A ℓ ρ = n ( q − q ,ρ = A s ρ = n ( q − q . Then the energy of C (attaining the bound (35) for every absolutely monotone h ) can be computedas E h ( C ) = M ( ρ h ( ℓ ) + ρ h ( s ))= q ( q + 1)( q − (cid:18) h (cid:18) − q q (cid:19) + h (cid:18) q − q q (cid:19)(cid:19) . Even more interesting example coming from [21] is given by an infinite series of codes con-structed by Dodunekov, Helleseth, and Zinoviev [19]. For a prime power q and positive integers m and ≤ N ≤ q m + 1 , the length and the cardinality of the codes from [19] are given by n = q m − q − N, | C | = q m . Further, for these codes we have s = − ( N − q m − N − q m − + NN ( q m − ,ℓ = − q m − q m − + 1 q m − , corresponding to d = ( N − q m − and D = N q m − , respectively. For these ℓ and s , the condition f ≥ for the polynomial f n,ℓ,s ( t ) is satisfied for N ≥ q m − q ) / (this corresponds to condition(16) from [21]). For such N and with the above n , ℓ and s we have | C | = q m = L ( n, ℓ, s ) . k = 1 (as in Example6.3). We have A ℓ = q m − − q ( q m − Nq − ,A s = q ( q m − Nq − . The h -energy is given by E h ( C ) = q m ( A ℓ h ( ℓ ) + A s h ( s )) and attains the bound (35) for every absolutely monotone h .Examples with k = 2 are already rare. In fact, there are many cases with integer L ( n, ℓ, s ) butmost of them (among the checked) fail to produce integer distance distributions from Theoremdd-codes. Two well known attaining codes are the projections of the binary Golay codes of lengths 23and 22. Indeed, the first projection of the binary Golay code has parameters n = 23 , ℓ = − / (i.e., D = 16 ), s = 7 / (i.e., d = 8 ) and | C | = 2 = L (23 , − ,
723 ) and the second projection has parameters n = 22 , ℓ = − / (i.e., D = 16 ), s = 3 / (i.e., d = 8 )and | C | = 2 = L (22 , − ,
311 ) . Acknowledgments.
The first author was partially supported by the National Scientific Program"Information and Communication Technologies for a Single Digital Market in Science, Educationand Security (ICTinSES)", financed by the Bulgarian Ministry of Education and Science. Thesecond author was supported, in part, by the Simons Foundation under CGM
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