Completely regular codes by concatenating Hamming codes
aa r X i v : . [ m a t h . C O ] M a r Completely regular codes by concatenatingHamming codes
J. Borges, J. Rif`a ∗ Department of Information and Communications Engineering,Universitat Aut`onoma de Barcelona,V. A. Zinoviev † A.A. Kharkevich Institute for Problems of Information Transmission,Russian Academy of SciencesOctober 4, 2018
Abstract
We construct new families of completely regular codes by con-catenation methods. By combining parity check matrices of cyclicHamming codes, we obtain families of completely regular codes. In allcases, we compute the intersection array of these codes. We also studywhen the extension of these codes gives completely regular codes.Some of these new codes are completely transitive. ∗ email: [email protected], [email protected] † e-mail: [email protected] Introduction
Let F q be a finite field of the order q . A q -ary linear [ n, k, d ; ρ ] q -code C is a k -dimensional subspace of F nq , where n is the length , d is the minimumdistance , q k is the cardinality of C , and ρ is the covering radius . For q = 2, weomit the subscript q . The packing radius of C is e = ⌊ ( d − / ⌋ . Given anyvector v ∈ F nq , its distance to the code C is d ( v , C ) = min x ∈ C { d ( v , x ) } andthe covering radius of the code C is ρ = max v ∈ F nq { d ( v , C ) } . Note that e ≤ ρ .We denote by D = C + x a coset of C , where + means the component-wiseaddition in F q .For a given q -ary code C of length n and covering radius ρ , define C ( i ) = { x ∈ F nq : d ( x , C ) = i } , i = 0 , , . . . , ρ. The sets C (0) = C, C (1) , . . . , C ( ρ ) are called the subconstituents of C .Say that two vectors x and y are neighbors if d ( x , y ) = 1. Denote by the all-zero vector. Definition 1.1 ([15]) . A q -ary code C of length n and covering radius ρ is completely regular , if for all l ≥ x ∈ C ( l ) has the samenumber c l of neighbors in C ( l −
1) and the same number b l of neighbors in C ( l + 1). Define a l = ( q − · n − b l − c l and set c = b ρ = 0. The parameters a l , b l and c l (0 ≤ l ≤ ρ ) are called intersection numbers and the sequence { b , . . . , b ρ − ; c , . . . , c ρ } is called the intersection array (shortly IA) of C .Let M be a monomial matrix, i.e. a matrix with exactly one nonzeroentry in each row and column. If q is prime, then the automorphism groupof C , Aut( C ), consists of all monomial ( n × n )-matrices M over F q such that c M ∈ C for all c ∈ C . If q is a power of a prime number, then Aut( C ) alsocontains any field automorphism of F q which preserves C . The group Aut( C )2cts on the set of cosets of C in the following way: for all π ∈ Aut( C ) andfor every vector v ∈ F nq we have π ( v + C ) = π ( v ) + C . Definition 1.2 ([9, 17]) . Let C be a linear code over F q with covering radius ρ . Then C is completely transitive if Aut( C ) has ρ + 1 orbits when acts onthe cosets of C .Since two cosets in the same orbit have the same weight distribution, itis clear that any completely transitive code is completely regular.Completely regular and completely transitive codes are classical subjectsin algebraic coding theory, which are closely connected with graph theory,combinatorial designs and algebraic combinatorics. Existence, constructionand enumeration of all such codes are open hard problems (see [6, 12, 15, 18]and references there).It is well known that new completely regular codes can be obtained bydirect sum of perfect codes or, more general, by direct sum of completelyregular codes with covering radius 1 [2, 17]. In the current paper, we extendthese constructions, giving several explicit constructions of new completelyregular and completely transitive codes, based on concatenation methods. In this section we see several results we will need in the next sections.
Lemma 2.1 ([15]) . Let C be a completely regular code with covering radius ρ and intersection array { b , . . . , b ρ − ; c , . . . , c ρ } . If C ( i ) and C ( i + 1) , ≤ i < ρ , are two subconstituents of C , then b i | C ( i ) | = c i +1 | C ( i + 1) | . efinition 2.2 ([10]) . A quasi-perfect e -error-correcting q -ary code C iscalled uniformly packed if there exist natural numbers λ and µ such that forany vector x : B x,e +1 = λ if d ( x, C ) = e,µ if d ( x, C ) = e + 1 . Van Tilborg [19] (see also [13, 16]) showed that no nontrivial codes of thiskind exist for e > Proposition 1 ([10], see also [16]) . A uniformly packed code is completelyregular.
Definition 2.3. A t -( v, k, λ )- design is an incidence structure ( S, B ), where S is a v -set of elements (called points ) and B is a collection of k -subsets ofpoints (called blocks ) such that every t -subset of points is contained in exactly λ > < t ≤ k ≤ v ).In terms of incident matrix a t -( v, k, λ )-design is a binary code C of length v with codewords of weight k such that any binary vector of length v andweight t is covered by exactly λ codewords. A t -design with λ = 1 is calleda Steiner system and also denoted by S ( v, k, t ). The following properties arewell known (e.g., [3, 4, 11]). Proposition 2.
Given a t - ( v, k, λ ) -design, every i -subset of points ( ≤ i ≤ t ) is contained in exactly λ i blocks, where λ i = λ (cid:0) v − it − i (cid:1)(cid:0) k − it − i (cid:1) . Corollary 1.
Given a t - ( v, k, λ ) -design D :(i) D is an i -design, for all i ≤ t .(ii) λ = λ t . iii) The number of blocks of D is b = λ .(iv) Each point is contained in the same number of blocks, namely r = λ = bk/v ( r is called the replication number ). There is a natural q -ary generalization of such t -designs (see [1, 8, 10, 20]).Let E = { , , . . . , q − } . A collection B of b vectors x , . . . , x b of length v and weight k over E is called a q -ary t -design and denoted t -( v, k, λ ) q , if forevery vector y over E of length v and weight t there are exactly λ vectors x i , . . . , x i λ from B such that d ( y, x i j ) = k − t for all j = 1 , . . . , λ . If λ = 1,then we obtain a q -ary Steiner system, denoted S ( v, k, t ) q .For a code C denote by C w the set of all codewords of C of weight w .Regularity of a code C implies that the sets C w determine t -designs.Directly from the definition of completely regular codes (see also [10, 16])we have the following Theorem 2.4.
Let C be a q -ary completely regular code of length n withminimum distance d .(i) If d = 2 e + 1 then any nonempty set C w is an e - ( n, w, λ w ) q -design.(ii) If d = 2 e + 2 then any nonempty set C w is an ( e + 1) - ( n, w, λ w ) q -design. For a code C , we denote by s + 1 the number of nonzero terms in the dualdistance distribution of C , obtained by the MacWilliams transform. Theparameter s was called external distance by Delsarte [8], and is equal to thenumber of nonzero weights of C ⊥ if C is linear. The following propertiesshow the importance of this parameter. Theorem 2.5. If C is any code with covering radius ρ and external distance s , then i) [8] ρ ≤ s .(ii) [8] A code C is perfect ( e = ρ ) if and only if e = s .(iii) [10] A code C is quasi-perfect uniformly packed if and only if s = e + 1 .(iv) [17] If C is completely regular, then ρ = s .(v) [8] If d ≥ s − , then C is completely regular.(vi) [6] If C has only even weights and d ≥ s − , then C is completelyregular. Given a code C , we define the extended code C ∗ by adding an extracoordinate to each codeword of C such that the sum of the coordinates ofthe extended vector is zero. Proposition 3.
If a binary extended code C ∗ , of length n + 1 , is a completelyregular code with minimum distance d ∗ = 2 e + 2 ≥ , then for all odd w | C ∗ w +1 | ( w + 1) = ( n + 1) | C w | and ( n − w ) | C w | = ( w + 1) | C w +1 | . Proof.
Let w be odd and assume that C w is not empty. By Theorem 2.4, theset C ∗ w +1 of codewords of weight w + 1 form a ( e + 1)-( n + 1 , w + 1 , λ ∗ )-designwhich, in particular, is a 2-( n + 1 , w + 1 , λ ∗ )-design, by Corollary 1. Thenumber of codewords in C ∗ w +1 with nonzero value at position n + 1 is r ∗ , thereplication number, and clearly r ∗ = | C w | . Therefore, | C ∗ w +1 | ( w + 1) = ( n + 1) r ∗ . (1)Combining (1) with | C ∗ w +1 | = | C w +1 | + | C w | , the result follows.6or any vector x = ( x , . . . , x n ) ∈ F nq , denote by σ ( x ) the right cyclic shiftof x , i.e. σ ( x ) = ( x n , x , . . . , x n − ). Define recursively σ i ( x ) = σ ( σ i − ( x )),for i = 2 , , . . . and σ ( x ) = σ ( x ). For j <
0, we define σ j ( x ) = σ ℓ ( x ), where ℓ = j mod n .Finally, we will also make use of the following technical lemma. Lemma 2.6.
Let x ∈ F nq be a vector of weight w . If gcd( n, w ) = 1 , then σ i ( x ) = x , for all i = 1 , . . . , n − .Proof. Assume that σ i ( x ) = x for some i = 2 , . . . , n −
1. Then, i divides n and x has the form: x = ( x ′ , x ′ , . . . , x ′ ) | {z } n/i , where x ′ is a vector of length i . Thus, w is a multiple of i . As a consequence, i is a common divisor of n and w . For the case, σ ( x ) = x , note that x shouldbe either the all-one or the all-zero vector. The next construction is new, although the dual codes of the resulting familyof q -ary completely regular codes are known as the family SU2 in [7]. In thecurrent paper, we also study when these codes are completely transitive andwhen the extended codes are completely regular. Construction I
Let H be the parity check matrix of a q -ary cyclic Hamming code of length n = ( q k − / ( q − n, q −
1) = 1). Thus, the simplex codegenerated by H is also a cyclic code. Denote by r , . . . , r k the rows of H .7or any c ∈ { , . . . , n } , consider the code C with parity check matrix H H . . . HH H . . . H c , (2)where H i is the matrix H after cyclically shifting i times its columns to theright. In other words, the rows of H i are σ i ( r ) , . . . , σ i ( r k ). Note that, for c = 1, we have HH , which generates the simplex code as H . Therefore, in this case, C is aHamming code. Proposition 4.
The code C ⊥ has nonzero weights w = cq k − and w = ( c − q k − . Proof.
Let x = ( x , . . . , x c ) ∈ C ⊥ be a nonzero codeword such that each x i is an vector of length n generated by HH i . Since H and H i generate the same simplex code, x i has weight 0 or q k − . As-sume that x i is the zero vector. Then, x i is generated by a linear combinationof the rows of H (giving some vector v ), together with a linear combinationof the rows of H i (giving the vector − v ). The same linear combination of therows of H j gives the vector u = σ j − i ( − v ). Since the weight of − v is q k − andgcd(( q k − / ( q − , q k − ) = 1, we have, by Lemma 2.6, u = − v and hence x j is not the zero vector.The conclusion is that x has weight cq k − or ( c − q k − .8 emark 1. In the proof of Proposition 4, the number of ways to get x i equal to the zero vector (being x a nonzero codeword) is equal to the numberof nonzero vectors generated by H . Therefore, C ⊥ has c ( q k −
1) codewordsof weight w = ( c − q k − and q k − c ( q k − − w = cq k − . By using this weight distribution of C ⊥ and the MacWilliamstransform [14], it is possible to compute | C | , the number of codewords in C of weight 3. Here we use a combinatorial argument to compute | C | .Let B , . . . , B c be the n -sets, which we call blocks , of coordinate positionscorresponding to H , . . . , H c , that is B j = { ( j − n + 1 , ( j − n + 2 , . . . , jn } ,for j = 1 , . . . , c . Proposition 5.
The number | C | of codewords in C of weight is: ( q − c (cid:0) n (cid:1) q − n (cid:18) c (cid:19) = ( q − cn q − n −
1) + ( c − c − , provided that (cid:0) c (cid:1) = 0 , for c ∈ { , } .Proof. For c = 1, the result is trivial since( q − (cid:0) n (cid:1) q − n ( n − q -ary 2-( n, , x of weight 3 cannot have exactly 2 nonzero coordinates in the same blockbecause there exists a codeword y in such block covering these two coordi-nates and, hence, we would have d ( x, y ) = 2. Thus, the result is also trivialfor c = 2.If c >
2, then the codewords of weight 3 are divided into two classes: a)those with the three nonzero coordinates in the same block, and b) thosewith the three nonzero coordinates in three different blocks.Clearly, the number of codewords in the case a) is c ( q − (cid:0) n (cid:1) . B j , B j , and B j (wecan choose these three blocks in (cid:0) c (cid:1) ways). In the block B j , we fix a vector v of weight one (we have ( q − n such vectors). Now, we claim that thereexists exactly one codeword of weight 3 covering v with the other two nonzerocoordinates in B j and B j .If there are two such codewords, say x = v + e + e and y = v + d + d ( e ℓ and d ℓ are one-weight vectors with the nonzero coordinate in B j ℓ , for ℓ = 2 , x ′ and y ′ withnonzero coordinates in B j and B j , respectively, and covering e + e and d + d , respectively. Therefore, x + y + x ′ + y ′ has weight 2 leading to acontradiction.If there are not any codeword covering v with nonzero coordinates in B j and B j , then any vector v + e is at distance two from C . Thus, wecan get ( q − n vectors z , such that d ( z, C ) = d ( z, ). We know that | C (2) | = ( q k − ( q − nc − | C | , since the covering radius of C is ρ = 2, andclearly | C (1) | = ( q − nc | C | . Therefore, the number of vectors in C (2) atdistance 2 from the zero codeword is ( q k − ( q − nc − q k − ( q − nc − ≥ ( q − n , which gives ( q k + 1)( q k − − ( q k − c ≥ ( q k − , and hence c ≤
2, whichcontradicts the assumption c >
Corollary 2.
The code C with parity check matrix given in (2) is a quasi-perfect uniformly packed code (hence completely regular) with parameters [ nc, nc − k,
3; 2] q and intersection array IA = { ( q − nc, (( q − n − c + 2) ( c − , c ( c − } . roof. The length, dimension and minimum distance of C are clear. ByProposition 4, C has external distance s = 2. Since C is not perfect, 1 < ρ .Thus, by Theorem 2.5 (i), the covering radius is ρ = 2, and by Theorem 2.5(iii), C is a quasi-perfect uniformly packed code.The values of the intersection numbers b = ( q − n and c = 1 arestraightforward since C has minimum distance 3.Now, we compute the intersection number a , that is, the number ofneighbors in C (1) of any vector z ∈ C (1). Without loss of generality, assumethat z is a one-weight vector. Then, a is the addition of the number oftwo-weight vectors covering z and covered by some codeword of weight 3,and the q − z . Since the set C ofcodewords of weight 3 defines a q -ary 1-design (Theorem 2.4), we have that3 | C | = ( q − cnr, (3)where r is the replication number, i.e. the number of codewords in C cover-ing z (note that (3) is a generalization to the q -ary case of Corollary 1 (iv)).Of course, any such codeword covers two vectors of weight 2 that, also, cover z . Thus, we have that a = 2 r + q −
2. Combining with (3), we obtain a = 6 | C | ( q − cn + q − , and substituting | C | from Proposition 5, we get a = [( q − n −
1) + ( c − c − q − . Since b = ( q − cn − c − a , we obtain b = ( q − cn − − [( q − n −
1) + ( c − c − − ( q − q − n − c + 2)( c − .
11y Lemma 2.1, we have that b | C (1) | = c | C (2) | . Since C has minimumdistance 3, we have | C (1) | = ( q − cn | C | . Also, | C (2) | = q cn − | C (1) | − | C | because the covering radius of C is ρ = 2. Therefore, we can compute c : c = b | C (1) || C (2) | = (( q − n − c + 2)( c − q − cn | C | ( q k − ( q − nc − | C | . Substituting n = ( q k − / ( q − c ( c − Remark 2.
Almost all codes in the family described in Construction I arenot completely transitive codes. However, software computations suggestthat in the binary case and for any value of k (so n = 2 k − c ∈ { , , n − , n } . In general,in the q -ary case when q is a power of two, the completely transitive codes arethose with c ∈ { , } and if q = p r , for p = 2, then the completely transitivecodes are those with c = 2. Remark 3.
By extending the codes in the family given in ConstructionI we do not obtain completely regular codes, except for the binary casewhen the parameter c equals 2 k − + 1. In this case, the family of extended[ n (2 k − + 1) + 1 , n (2 k − + 1) − k,
4; 3] codes we obtain coincides with thefamily described in Theorem 3.1.
Construction II
The next construction works again for q -ary cyclic Hamming [ n, k,
3; 1] q codes, where n = ( q k − / ( q −
1) and gcd( n, q −
1) = 1. For a givensuch code of length n with parity check matrix H , the matrices H i , i =1 , , . . . , n − c be any integer from therange: 1 ≤ c ≤ n − C be the code with parity check matrix12 ( c ) = H H H . . . H H H H . . . H c where 0 denotes the zero matrix (of the same size as H ). Proposition 6.
The code C ⊥ has nonzero weights w = ( c + 3) q k − and w = ( c + 2) q k − , except if c = n − and q = 2 . In this case, C ⊥ has only the nonzero weight w = 2 k − and C is a Hamming code of length k − .Proof. Assume that c < n − q >
2. As in Construction I, let B , . . . , B c +3 be the sets (blocks) of consecutive n coordinate positions. Note that any lin-ear combination of the first (respectively, second) k rows of H ( c ) gives acodeword in C ⊥ with the zero vector in, and only in, the block B (respec-tively, B ). For a linear combination which gives nonzero vectors in B and B , we have that the obtained codeword in C ⊥ can have the zero vector inat most one block B j , for j = 3 , . . . , c . This is true by the same argumentused in the proof of Proposition 4.In the binary case, if c = n −
1, we have that any nonzero codeword in C ⊥ has the zero vector in exactly one block. Indeed, any linear combinationwhich gives nonzero vectors in B and B , gives some vector v + u , where v is generated by the first k rows of H ( c ) , and u is generated by the second k rows of H ( c ) . Clearly, v and u have the forms: v = ( y, , y, y, . . . , y ) , and u = (cid:0) , x, x, σ ( x ) , σ ( x ) , . . . , σ n − ( x ) (cid:1) , for some x, y ∈ F nq . Since x, σ ( x ) , σ ( x ) , . . . , σ n − ( x ) are all different byLemma 2.6, and a simplex code of length n contains n nonzero codewords,13e conclude that y ∈ { x, σ ( x ) , σ ( x ) , . . . , σ n − ( x ) } . Therefore, C ⊥ has only the weight w = ( c + 2)2 k − = 2 k − . In this case, C has length ( c + 3) n = (2 k + 1)(2 k −
1) = 2 k −
1. Since the minimum distanceis 3 and the dimension is 2 k , C is a Hamming code. Proposition 7.
The number of codewords in C of weight is: | C | = ( c + 3) n ( q − n − q −
1) + ( c + 1)( c + 2)] . Proof.
We compute separately the number of codewords in C for the differ-ent possible cases.a) Codewords in C with the three nonzero coordinates in B ∪ · · · ∪ B c +3 .We can apply here the arguments of Proposition 5 for c + 1 instead of c . The result is:( c + 1) n ( q − q − n −
1) + c ( c − . (4)b) Codewords in C with the three nonzero coordinates in B ∪ B . Clearly,all the nonzero coordinates must be in B or in B . Since the triplesin B (or B ) form a Steiner system (Theorem 2.4), we have that thisnumber of codewords is:2 ( q − n ( n − q − n ( n − . (5)c) Codewords in C with exactly one nonzero coordinate in B ∪· · ·∪ B c +3 .Consider any column h i of H ( c ) in B ∪ · · · ∪ B c +3 . It is clear that thereis exactly one column h j in B and one column h ℓ in B , such that h i , h j and h ℓ are linearly dependent. Hence, in this case we have exactly14ne codeword for each coordinate (and its multiples) in B ∪ · · · ∪ B c +3 .Thus, the result is: ( q − n ( c + 1) . (6)d) Codewords in C with exactly one nonzero coordinate in B ∪ B . Theremaining pair of nonzero coordinates cannot be in the same block.Indeed, one such pair of coordinates is already covered by a triple inthe same block. The corresponding columns of H ( c ) must have equal(up to multiples) the first k coordinates or the second k coordinates(depending on the given nonzero coordinate is either in B or in B ,respectively). Hence, for any pair of blocks in { B , . . . , B c +3 } we canchoose n columns (and their q − n ( q − (cid:18) c + 12 (cid:19) = ( c + 1) cn ( q − . (7)Adding (4), (5), (6), and (7), we obtain the statement. Corollary 3. (i) For q = 2 and c = n − , C is a binary Hamming code of length k − .(ii) For q > or c < n − , C is a linear completely regular [( c + 3) n, ( c +3) n − k,
3; 2] q code with intersection array IA = { ( c + 3) n ( q − , ( c + 2) (( q − n − − c ) ; 1 , ( c + 2)( c + 3) } . Proof. (i) It is already proved in Proposition 6.(ii) The length, dimension and minimum distance of C are clear. ByProposition 6, C has external distance s = 2. Since C is not perfect, we havethat ρ > ρ = 2. Hence, byTheorem 2.5 (iii), C is a quasi-perfect uniformly packed code.15he values of the intersection numbers b = ( c + 3) n ( q −
1) and c = 1are straightforward since C has minimum distance 3.Now, we compute the intersection number a , that is, the number ofneighbors in C (1) of any vector z ∈ C (1). Without loss of generality, assumethat z is a one-weight vector. Then, a is the addition of the number oftwo-weight vectors covering z and covered by some codeword of weight 3,and the q − z . Since the set C ofcodewords of weight 3 defines a q -ary 1-design (Theorem 2.4), we have that3 | C | = ( q − cnr, (8)where r is the replication number, i.e. the number of codewords in C cov-ering z . Of course, any such codeword covers two vectors of weight 2 that,also, cover z . Thus, we have that a = 2 r + q −
2. Combining with (8), weobtain a = 6 | C | ( c + 3) n ( q −
1) + q − , and substituting | C | from Proposition 7, we get a = ( q − n − c + 1)( c + 2) . Since b = ( c + 3)( q − n − c − a , we obtain b = ( c +3)( q − n − − [( q − n − c + 1)( c + 2)] = ( c +2)(( q − n − − c ) . By Lemma 2.1, b | C (1) | = c | C (2) | . Since C has minimum distance 3, wehave | C (1) | = ( c + 3)( q − n | C | . Also, | C (2) | = q ( c +3) n − | C (1) | − | C | becausethe covering radius of C is ρ = 2. Therefore, we can compute c : c = b | C (1) || C (2) | = ( c + 2)(( q − n − − c )( c + 3)( q − n | C | ( q k − ( c + 3)( q − n − | C | . Substituting n = ( q k − / ( q − c +2)( c +3).16 emark 4. Almost all codes in the family given in Corollary 3 are not com-pletely transitive. However, in the binary case, using computer calculations,we conjecture that for any value of k >
2, the codes of Corollary 3, arecompletely transitive for c ∈ { k − , k − , k − } . Remark 5.
In the binary case, the extension of the codes given in Corollary3 are not completely regular in almost all cases. However, for each value of k , there are exactly two values of c such that the obtained binary extendedcode is completely regular as we show in the next theorem.Of course, for q = 2 and c = n −
1, the extended code is an extendedHamming code. Therefore, we consider the binary cases where 1 ≤ c ≤ n − Theorem 3.1.
For ≤ c ≤ n − and q = 2 , the extended code C ∗ is acompletely regular code if and only if c = 2 k − − . In this case, C is a [2 k − (2 k + 1) , k − (2 k + 1) − − k,
4; 3] code with
IA = { k − (2 k + 1) , k − (2 k + 1) − , k − ; 1 , k − (2 k − + 1) , k − (2 k + 1) } . Proof.
As can be seen in [5, Prop. 1.1], C ∗ has covering radius ρ ∗ = ρ +1 = 3.Hence, if C ∗ is completely regular, it must have external distance s ∗ = 3. Inother words, ( C ∗ ) ⊥ must have exactly 3 nonzero weights (Theorem 2.5 (iv)).A generator matrix for ( C ∗ ) ⊥ is obtained adding, first the zero column, andsecond the all-one row to the matrix H ( c ) . Therefore, ( C ∗ ) ⊥ has at least thenonzero weights w = ( c + 3) n + 1, w = ( c + 3)2 k − and w = ( c + 2)2 k − (see Proposition 6). If ( C ∗ ) ⊥ has exactly these weights, then it is clear that w + w = w . This condition is equivalent to(2 c + 5)2 k − = ( c + 3)(2 k −
1) + 1 , which implies c = 2 k − −
2. 17n that case, C ∗ is a [2 k − (2 k + 1) , k − (2 k + 1) − − k,
4; 3] code and,by Theorem 2.5 (vi), we have that C ∗ is a completely regular code.Now, we compute the intersection numbers. Let N = 2 k − (2 k + 1) bethe length of C ∗ . Since the minimum distance in C ∗ is 4, it is clear that b = N and c = 1. Moreover, giving a one-weight vector z ∈ C ∗ (1), all itsneighbors, except the zero codeword, are vectors in C ∗ (2). Thus, b = N − y ∈ C ∗ (3), we have that all its neighbors are in C ∗ (2). Hence, c = N .By Proposition 3, we have that 4 | C ∗ | = N | C | . From this and Proposition7, we obtain: | C ∗ | = 2 k − (2 k + 1)(2 k − + 1)(2 k − · (cid:2) k − k − (2 k − − (cid:3) = 2 k − (2 k − − k − k − + 1)12 . (9)By Theorem 2.4, the set C ∗ defines a 2-( N, , λ )-design. Using Proposition2 and (9), we can compute the parameter λ : | C ∗ | = λ N ( N − ⇒ λ = 2 k − (2 k − − k − k − + 1)2 k − (2 k + 1) (2 k − (2 k + 1) − k − − k − k − + 1)2 k − (2 k + 1) − . (10)Let y ∈ C ∗ (2), without loss of generality, we can assume that y has weight2. Then, y is at distance two of exactly λ + 1 codewords in C ∗ . Since anycodeword has (cid:0) N (cid:1) vectors at distance 2, and all such vectors are in C ∗ (2),we have the relation: | C ∗ | N ( N − λ + 1) | C ∗ (2) | . (11)Alternatively, (11) can be obtained counting in two ways the number of edgesof the bipartite graph that has C ∗ ∪ C ∗ (2) as set of vertices, and a vertex18n C ∗ is adjacent to a vertex in C ∗ (2) if the corresponding vectors are atdistance two. Now, using (10) and (11), we can compute | C ∗ (2) | : | C ∗ (2) | = | C ∗ | k − (2 k + 1) (cid:2) k − (2 k + 1) − (cid:3) (cid:2) k − (2 k + 1) − (cid:3) (2 k − − k − k − + 1) + 2 k − (2 k + 1) − | C ∗ | k − (2 k + 1)(2 k − + 2 k − − k + 2 k − − k + 1)2 k − (2 k − − k − + 2 k − | C ∗ | (2 k + 1)(2 k −
1) = | C ∗ | (2 k − . (12)Next, we compute | C ∗ (3) | = 2 N − | C ∗ | − | C ∗ (1) | − | C ∗ (2) | . Clearly, | C ∗ (1) | = N | C ∗ | . Therefore, using (12), we obtain: | C ∗ (3) | = | C ∗ | (cid:2) k +1 − − k − (2 k + 1) − (2 k − (cid:3) = | C ∗ | k − (2 k − . (13)By Lemma 2.1, we have that b | C ∗ (1) | = c | C ∗ (2) | and b | C ∗ (2) | = c | C ∗ (3) | . Using these relations, (12), and (13), we obtain: c = N ( N − k + 1)(2 k −
1) = 2 k − (2 k + 1)(2 k − + 2 k − − k + 1)(2 k −
1) = 2 k − (2 k − + 1); b = 2 k − (2 k + 1)2 k − (2 k − k + 1)(2 k −
1) = 2 k − . The statement is proved.
We have computationally checked that the following codes are completelyregular with the specified parameters.1. Let C be the binary [15 , ,
3; 3]-code with parity check matrix H = K K K K K K K K K , where K = K (respectively, K ) is obtained by one cyclic shift of the columnsof K in one position (respectively, by two cyclic shifts). Then, C is a[15 , ,
3; 3] completely regular code with IA = { , ,
1; 1 , , } . Thebinary [16 , ,
4; 4]-code, obtained by extension of C , is completely reg-ular with IA = { , , ,
1; 1 , , , } .
2. Denote by D ( u, q ) a difference matrix [3], i.e. a square matrix of the or-der qu over an additive group of order q , such that the component-wisedifference of any two rows contains any element of the group exactly u times.Take the difference matrix D (2 , D = Let H be a binary (12 ×
18) matrix obtained from D by changing anyelement i by the matrix K i . Then, the [18 , ,
3; 2] code with paritycheck matrix H is a completely regular code with IA = { ,
15; 1 , } .
3. Do the same construction as in item 2 for the matrix D ∗ , which isthe difference matrix D (2 ,
3) without the trivial column. The result-ing [15 , ,
3; 3] code is CR with IA = { , ,
1; 1 , , } . This codecoincides with the code described in item 1.20 cknowledgements This work has been partially supported by the Spanish grants TIN2016-77918-P, AEI/FEDER, UE., MTM2015-69138-REDT; the Catalan AGAURgrant 2014SGR-691 and also by Russian fund of fundamental researches (15-01-08051).
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