Generalized Hamming weights of toric codes over hypersimplices and square-free affine evaluation codes
aa r X i v : . [ m a t h . A C ] O c t GENERALIZED HAMMING WEIGHTS OF TORIC CODES OVERHYPERSIMPLICES AND SQUARE-FREE AFFINE EVALUATIONCODES
NUPUR PATANKER AND SANJAY KUMAR SINGH
Abstract.
Let F q be a finite field with q elements, where q is a power of prime p . Apolynomial over F q is square-free if all its monomials are square-free. In this note, wedetermine an upper bound on the number of zeroes in the affine torus T = ( F ∗ q ) s ofany set of r linearly independent square-free polynomials over F q in s variables, undercertain conditions on r , s and degree of these polynomials. Applying the results, wepartly obtain the generalized Hamming weights of toric codes over hypersimplices andsquare-free evaluation codes, as defined in [1]. Finally, we obtain the dual of these toriccodes with respect to the Euclidean scalar product. Introduction
The fundamental parameters of linear codes, such as dimension and minimum distance,determine the efficiency and error-correction capability of the codes. Another importantproperty of linear codes is their generalized Hamming weights. The notion of generalizedHamming weights for a linear code C over F q is defined as follows.For any F q -subspace D of [ n, k ] code C , the support of D is defined as supp ( D ) := { ≤ i ≤ n : x i = 0 for some x = ( x , · · · , x n ) ∈ D } . For 1 ≤ r ≤ k , the r -th generalized Hamming weight of C is defined as d r ( C ) := min { | supp ( D ) | : D is a linear subcode of C with dim ( D ) = r } . In particular, the first generalized Hamming weight of C is the usual minimum distance.The set of generalized Hamming weights { d ( C ) , d ( C ) , · · · , d k ( C ) } is called the weighthierarchy of code C . The notions of generalized Hamming weights for linear codes wereintroduced in [2], [10], and rediscovered by Wei in his paper [3]. These weights completelycharacterize the performance of the code on the wire-tap channel of type II, and also theperformance as a t -resilient function. The generalized Hamming weights of various linearcodes have been studied for many years.Toric codes were introduced by J. Hansen in [13] and since then have been studied in[14], [15], [16], [17], [18], [19], [20], [21], etc. Projective Reed-Muller-type code over theprojective torus has been studied in [22], [23], etc. Recently, Delio Jaramillo, Maria Vaz Mathematics Subject Classification.
Pinto and Rafael H. Villarreal, in [1], introduced affine and projective toric codes overhypersimplices. The authors computed their dimension and minimum distance. Theyalso introduced square-free evaluation codes and computed their dimension, minimumdistance and second generalized Hamming weight. They posed the problem of obtainingformulae for the generalized Hamming weights of these codes. In this note, we determinethe generalized Hamming weights of toric codes over hypersimplices and square-free affineevaluation code.The problem of finding the generalized Hamming weights of toric codes over hypersim-plices can be solved by answering the following question stated in terms of polynomials:Let s and d be integers such that s ≥ ≤ d ≤ s . For 1 ≤ r ≤ (cid:0) sd (cid:1) , let f , f , · · · , f r be linearly independent homogeneous square-free polynomials of degree d in s variables with coefficients in F q . What is the maximum number of solutions in affinetorus T = ( F ∗ q ) s of the system f = f = · · · = f r = 0?In [1], the answer to this problem is given for r = 1. Our goal in this note is tosolve a more generalized problem where f , f , · · · , f r are linearly independent square-free polynomials of degree d in s variables with coefficients in F q . To obtain our results, wefollow the footsteps of [6]. Another related question is to solve the above-stated problemwhen f , f , · · · , f r are linearly independent square-free polynomials of degree at most d in s variables with coefficients in F q , where 1 ≤ r ≤ P di =0 (cid:0) si (cid:1) . The answer to this problemhelps us to determine the generalized Hamming weights of square-free affine evaluationcodes. The answer for r = 1 , d + r − < s and as an application, determine the generalized Hammingweights of these codes.This note is organized as follows. In section 2, we recall the definition of toric code overhypersimplices and square-free evaluation codes, as defined in [1]. We also study the affineHilbert function. In section 3, we determine an upper bound on the number of solutionsin the affine torus of any set of r linearly independent square-free polynomials over F q ofdegree d in s variables, 1 ≤ r ≤ (cid:0) sd (cid:1) . We also determine an upper bound on the numberof solutions in affine torus of any set of r linearly independent square-free polynomialsover F q of degree at most d in s variables, 1 ≤ r ≤ P di =0 (cid:0) si (cid:1) . In section 4, we determinethe generalized Hamming weights of the toric codes over hypersimplices and square-freeevaluation codes in specific cases. In section 5, we conclude the note by determining thedual of toric codes over hypersimplices with respect to the Euclidean scalar product.2. Preliminaries
Let s and d be integers such that s ≥ ≤ d ≤ s . In this section, we recall thedefinitions of toric codes over hypersimplices and square-free affine evaluation codes. Wealso recall the known results on these codes and study the affine Hilbert function.Throughout this note, we use the notation K := F q , where q is a power of prime p . ENERALIZED HAMMING WEIGHTS OF CODES 3
Evaluation codes over d -th hypersimplex, [1] . Let S := K [ t , · · · , t s ] = L ∞ d =0 S d be the polynomial ring in s variables over K with standard grading.Let P be the convex hull in R s of all integral points e i + e i + · · · + e i d such that1 ≤ i < · · · < i d ≤ s , where e i is the i -th unit vector in R s . The lattice polytope P is called the d -th hypersimplex in R s . The affine torus of the affine space A s is givenby T := ( K ∗ ) s , where K ∗ is the multiplicative group of K . The projective torus of theprojective space P s − over K is given by T := [ T ], where [ T ] is the image of T underthe map φ : A s \{ } → P s − , a [ a ]. The cardinality of T is m := ( q − s and thecardinality of T is ¯ m := ( q − s − . Let V d be the set all monomials t a := t a t a · · · t a s s such that a ∈ P ∩ Z s and let KV d be the vector space over K generated by V d . Thus, KV d is the space of homogeneous square-free polynomials of S of degree d . Denote by P , P , · · · , P m all points of the affine torus T of A s and denote by [ Q ] , [ Q ] , · · · , [ Q ¯ m ]all points of the projective torus T of P s − . We assume that the first entry of each Q i is1. Thus, T = { } × ( F ∗ q ) s − .The affine toric code C d of P of degree d is defined as the image of the evaluation map(2.1) ev d : KV d → F mq , ev d ( f ) := ( f ( P ) , f ( P ) , · · · , f ( P m )) . The code C d has length m . The minimum distance of C d is given by δ ( C d ) := min { | T \ V T ( f ) | : f ∈ KV d \ I ( T ) } , where V T ( f ) denotes the set of zeroes of f ∈ S in T .The projective toric code C P d of P of degree d is defined as the image of the evaluationmap(2.2) ev d : KV d → F ¯ mq , ev d ( g ) := ( g ( Q ) , g ( Q ) , · · · , g ( Q ¯ m )) . The code C P d has length ¯ m . The minimum distance of C P d is given by δ ( C P d ) := min { | T \ V T ( g ) | : g ∈ KV d \ I ( T ) } , where V T ( g ) denotes the set of zeroes of g ∈ S in T .The dimension and minimum distance of C d and C P d are given by the following theorems. Theorem 2.1. ([1] , Proposition . Let C d and C P d be the affine and projective toric codeof P of degree d , respectively. Then dim K ( C d ) = dim K ( C P d ) = ((cid:0) sd (cid:1) , if q ≥ , , if q = 2 . NUPUR PATANKER AND SANJAY KUMAR SINGH
Theorem 2.2. ([1] , Theorem . Let C d be the affine toric code of P of degree d and let δ ( C d ) be its minimum distance. Then δ ( C d ) = ( q − d ( q − s − d , if d ≤ s/ , q ≥ , ( q − s − d ( q − d , if s/ < d < s, q ≥ , ( q − s , if d = s, , if q = 2 . and let C P d be the projective toric code of P of degree d and let δ ( C P d ) be its minimumdistance. Then δ ( C P d ) = ( q − d ( q − s − d − , if d ≤ s/ , q ≥ , ( q − s − d ( q − d − , if s/ < d < s, q ≥ , ( q − s − , if d = s, , if q = 2 . Square-free affine evaluation code.
Let V ≤ d be the set of all square-free mono-mials of S of degree at most d and KV ≤ d be the corresponding subspace of S ≤ d . If wereplace KV d by KV ≤ d in the evaluation map of equation (2 . C ≤ d , is called a square-free affine evaluation code of degree d on T .The following results, proved in [1] give the dimension, minimum distance and secondgeneralized Hamming weight of C ≤ d . Proposition 2.3. ([1] , Proposition . Let C ≤ d be the square-free affine evaluation codeof degree d on the affine torus T = ( K ∗ ) s . Then, the length of C ≤ d is ( q − s , and thedimension of C ≤ d is given by dim K ( C ≤ d ) = ((cid:0) s (cid:1) + (cid:0) s (cid:1) + · · · + (cid:0) sd (cid:1) , if q ≥ , , if q = 2 . Theorem 2.4. ([1] , Theorem . If q ≥ , then the minimum distance δ ( C ≤ d ) of C ≤ d is ( q − d ( q − s − d . Theorem 2.5. ([1] , Theorem . If q ≥ and d ≥ , then the second generalizedHamming weight of C ≤ d is δ ( C ≤ d ) = ( ( q − s − ( q − , if d = s, ( q − d ( q − s − d − q, if d < s. Affine Hilbert function.
In this subsection, we briefly discuss the affine Hilbertfunction of an ideal I ⊂ K [ t , · · · , t s ]. For more details on this topic refer to [5] and [12].Let K [ t , · · · , t s ] ≤ u denotes the subset of K [ t , · · · , t s ] consisting of polynomials of totaldegree ≤ u . For an ideal I ⊂ K [ t , · · · , t s ], we denote by I ≤ u the subset of I consisting ofpolynomials of degree ≤ u . ENERALIZED HAMMING WEIGHTS OF CODES 5
Definition 2.6.
The affine Hilbert function of I is the function on the non-negativeintegers u defined by a HF I ( u ) := dim K K [ t , · · · , t s ] ≤ u /I ≤ u = dim K K [ t , · · · , t s ] ≤ u − dim K I ≤ u . Note that if I ⊆ J are any ideals of K [ t , · · · , t s ], then a HF I ( u ) ≥ a HF J ( u ). Given asubset X of K s , let I ( X ) denotes the vanishing ideal of X in K [ t , · · · , t s ]. Then theaffine Hilbert function of X , denoted by a HF X ( u ), is defined as a HF X ( u ) := a HF I(X) ( u ).We have the following result on the affine Hilbert function of an ideal of K [ t , · · · , t s ].The proof can be found in [5], Chapter 9, section 3. Proposition 2.7.
Fix a graded monomial ordering ≺ on K [ t , · · · , t s ] , then (1) For any ideal I of K [ t , · · · , t s ] , we have a HF I ( u ) = a HF h LT(I) i ( u ) . (2) If I is a monomial ideal of K [ t , · · · , t s ] , then a HF I ( u ) is the number of monomialsof degree at most u that does not lie in I . Another important result is the following proposition which can be found in [12], Lemma2 .
1. A similar statement can be found in [11], Corollary 4 . Proposition 2.8. ([6] , Proposition . Let Y ⊆ K s be a finite set. Then, | Y | = a HF Y ( u ) for sufficiently large u . Zeroes of square-free polynomials in the affine torus T = ( F ∗ q ) s ⊆ A s Throughout this section, we take ≺ to be the standard graded lexicographic order on S with t s ≺ · · · ≺ t ≺ t .For two distinct square-free polynomials f and g in S of degree d in s variables, thefollowing two lemmas give an upper bound on the cardinality of the sets V T ( f ) and V T ( f ) ∩ V T ( g ). Lemma 3 . .
3. We give another proofof the proposition. First, we need the following lemma from [1]. We add the proof for theconvenience of the reader.
Lemma 3.1.
Let h be a square-free polynomial in S \ F q . If h = ( t − α ) h for some α ∈ F ∗ q and h ∈ S , then h is a square-free polynomial in the variables t , t , · · · , t s .Proof. Let h = P wi =1 β i f i where β i ∈ F ∗ q , 1 ≤ i ≤ w and f , f , · · · , f w are distinctmonomials. Then(3.1) h = β t f + · · · + β w t f w − αβ f − · · · − αβ w f w . Assume that t divides f j for some 1 ≤ j ≤ w and choose j and n ≥ t n divides f j and t n +11 does not divides f i for i = 1 , · · · , w . As h is square-free, by equation (3 . t f j must be equal to f l for some 1 ≤ l ≤ w , a contradiction because t n +11 does not divides f l . This shows that h is a polynomial in the variables t , t , · · · , t s . NUPUR PATANKER AND SANJAY KUMAR SINGH
Hence t f , t f , · · · , t f w , f , f , · · · , f w are distinct monomials. As h is square-free, byequation (3 . f i is square-free for i = 1 , , · · · , w , i.e. h is square-free. (cid:3) Lemma 3.2.
Let s ≥ and ≤ d ≤ s . For any non-zero square-free polynomial g ofdegree d in F q [ t , t , · · · , t s ] , we have | V T ( g ) | ≤ ( q − s − ( q − d ( q − s − d . Proof.
We prove this lemma by induction on s . For s = 2, we have either d = 1 or d = 2.When d = 1, we have to show that | V T ( g ) | ≤ ( q − λ, µ, δ ∈ F ∗ q , g | V T ( g ) | λt + µ q − λt λt λt + µ q − λt + µt q − λt + µt + δ q − g in two variables t , t ofdegree one and the second column specifies the number of zeroes in T of the correspondingpolynomial. From the above table, we have | V T ( g ) | ≤ ( q − d = 2, we have to show that | V T ( g ) | ≤ q − . By direct calculations, we havethe following table, where λ, µ, δ, ρ ∈ F ∗ q , g | V T ( g ) | λt t λt t + µ q − λt t + µt q − λt t + µt + δ q − λt t + µt q − λt t + µt + δ q − λt t + µt + δt + ρ ≤ q − | V T ( g ) | ≤ (2 q − s = 2.Now, we assume that s ≥
3. We consider the following two cases: • If g ( α, t , · · · , t s ) = 0 for some α ∈ F ∗ q , then g = ( t − α ) h + h where no term of h is divisible by t . Putting t = α , we get that h is the zero polynomial. Thus, g = ( t − α ) h . If deg g = 1, then h ∈ F ∗ q and | V T ( g ) | = ( q − s − ≤ ( q − s − ( q − d ( q − s − d . ENERALIZED HAMMING WEIGHTS OF CODES 7
Therefore, we assume that deg g ≥
2. By Lemma 3 . h is square-free polynomialin t , t , · · · , t s and also we have deg h = d −
1. Let T ′ := ( F ∗ q ) s − . Then, byinduction hypothesis | V T ( g ) | = | V T ( t − α ) | + | V T ( h ) | − | V T ( t − α ) ∩ V T ( h ) | = ( q − s − + ( q − | V T ′ ( h ) |≤ ( q − s − + ( q − q − s − − ( q − d − ( q − s − d ]= ( q − s − + ( q − q − s − − ( q − d ( q − s − d = ( q − s − ( q − d ( q − s − d . • If g ( α, t , · · · , t s ) = 0 for any α ∈ F ∗ q , then let F ∗ q := { β , β , · · · , β q − } . For1 ≤ i ≤ q −
1, define g i ( t , t , · · · , t s ) := g ( β i , t , · · · , t s ). We have the followinginclusion V T ( g ) ֒ → ∪ q − i =1 ( { β i } × V T ′ ( g i )) , a a. Therefore | V T ( g ) | ≤ P q − i =1 | V T ′ ( g i ) | . For each i , 1 ≤ i ≤ q −
1, we have the followingcases.(1) If each term of degree d in g contains t , then g i is a square-free polynomialin s − d − d in g not containing t , then g i is a square-freepolynomial in s − d .Now, if each g i is of type (1), then | V T ( g ) | ≤ q − X i =1 | V T ′ ( g i ) |≤ ( q − q − s − − ( q − d − ( q − s − d ]= ( q − s − ( q − d − ( q − s − d +1 ≤ ( q − s − ( q − d ( q − s − d , as ( q − d − ( q − s − d +1 ≥ ( q − d ( q − s − d . But if there exists atleast one g i oftype (2), then using the fact that ( q − s − ( q − d ( q − s − d ≥ ( q − s − ( q − d ′ ( q − s − d ′ , for d > d ′ , we have | V T ( g ) | ≤ q − X i =1 | V T ′ ( g i ) |≤ ( q − q − s − − ( q − d ( q − s − d − ]= ( q − s − ( q − d ( q − s − d . (cid:3) NUPUR PATANKER AND SANJAY KUMAR SINGH
Lemma 3.3.
For s ≥ and d < s , let f and g be two distinct non-zero square-freepolynomials of degree d in F q [ t , t , · · · , t s ] . Then | V T ( f ) ∩ V T ( g ) | ≤ ( q − s − q ( q − d ( q − s − d − . Proof.
We prove this lemma by induction on s . When s = 2, d = 1 and we have to showthat for any two distinct square-free polynomials f and g in two variables of degree one,we have | V T ( f ) ∩ V T ( g ) | ≤
1. For λ, µ, δ, α, β ∈ F ∗ q , we obtain the following table by directcalculations. f g | V T ( f ) ∩ V T ( g ) | λt + µ αt λt + µ αt λt + µ αt + β λt + µ αt + βt λt + µ αt + βt + δ ≤ λt g λt g λt + µ αt + βt λt + µ αt + βt + δ ≤ λt + µt αt + βt + δ ≤ | V T ( f ) ∩ V T ( g ) | ≤
1. Thus, the lemma is true for s = 2. So,we assume that s ≥
3. Let T ′ := ( F ∗ q ) s − . To prove the lemma we consider the followingcases. • If f = ( t − α ) f and g = ( t − α ) g for some α ∈ F ∗ q . Note that if d = 1, then f and g are equal. So, we assume that d ≥
2. Now by induction hypothesis | V T ( f ) ∩ V T ( g ) | = | V T ( t − α ) | + | V T ( f ) ∩ V T ( g ) | − | V T ( t − α ) ∩ V T ( f ) ∩ V T ( g ) | = ( q − s − + ( q − | V T ′ ( f ) ∩ V T ′ ( g ) |≤ ( q − s − + ( q − q − s − − q ( q − d − ( q − s − d − ]= ( q − s − q ( q − d ( q − s − d − . • If f = ( t − α ) f and g = ( t − β ) g for some α, β ∈ F ∗ q with α = β . If d = 1,then f , g ∈ F ∗ q and | V T ( f ) ∩ V T ( g ) | = 0 ≤ ( q − s − q ( q − d ( q − s − d − . So we
ENERALIZED HAMMING WEIGHTS OF CODES 9 assume that d ≥
2, then by inclusion-exclusion principle | V T ( f ) ∩ V T ( g ) | = | ( V T ( t − α ) ∪ V T ( f )) ∩ ( V T ( t − β ) ∪ V T ( g )) | = | V T ( t − α ) ∩ V T ( g ) | + | V T ( t − β ) ∩ V T ( f ) | + | V T ( f ) ∩ V T ( g ) |− | V T ( t − α ) ∩ V T ( g ) ∩ V T ( f ) | − | V T ( t − β ) ∩ V T ( f ) ∩ V T ( g ) | = | V T ′ ( g ) | + | V T ′ ( f ) | + ( q − | V T ′ ( f ) ∩ V T ′ ( g ) |≤ q − s − − ( q − d − ( q − s − d ] + ( q − q − s − − q ( q − d − ( q − s − d − ]= ( q − s − ( q − d − ( q − s − d − ( q − q − ≤ ( q − s − q ( q − d ( q − s − d − , as ( q − d − ( q − s − d − ( q − q − ≥ q ( q − d ( q − s − d − . • If ( t − α ) ∤ f for any α ∈ F ∗ q but g = ( t − β ) g for some β ∈ F ∗ q , then let F ∗ q := { β , β , · · · , β q − } and for each i, 1 ≤ i ≤ q −
1, set f i ( t , t , · · · , t s ) := f ( β i , t , · · · t s ) and β = β j for some 1 ≤ j ≤ q −
1. Thus, we have | V T ( f ) ∩ V T ( g ) | ≤ | ( ∪ q − i =1 V T ′ ( f i )) ∩ V T (( t − β ) g ) | = | ( ∪ q − i =1 V T ′ ( f i )) ∩ V T ( t − β ) | + | ( ∪ q − i =1 V T ′ ( f i )) ∩ V T ( g ) |− | ( ∪ q − i =1 V T ′ ( f i )) ∩ V T ( t − β ) ∩ V T ( g ) | = | V T ′ ( f j ) | + q − X i =1 | V T ′ ( f i ) ∩ V T ′ ( g ) | − | V T ′ ( f j ) ∩ V T ( g ) | . ≤ | V T ′ ( f j ) | + ( q − | V T ′ ( g ) | . ≤ ( q − s − − ( q − d ( q − s − d − + ( q − q − s − − ( q − d − ( q − s − d ]= ( q − s − ( q − d ( q − s − d − [ q − q − s − q ( q − d ( q − s − d − . • If ( t − α ) ∤ f and ( t − α ) ∤ g for any α ∈ F ∗ q , then for 1 ≤ i ≤ q −
1, set f i ( t , · · · , t s ) := f ( β i , t , · · · , t s ) and g i ( t , · · · , t s ) := g ( β i , t , · · · , t s ). Thus, wehave(3.2) | V T ( f ) ∩ V T ( g ) | ≤ q − X i =1 | V T ′ ( f i ) ∪ V T ′ ( g i ) | . Now for each i , 1 ≤ i ≤ q −
1, we have the following cases.(1) If deg f i = deg g i = d − . If d = 1 then f i , g i ∈ F ∗ q and | V T ′ ( f i ) ∩ V T ′ ( g i ) | = 0.So, we assume d ≥
2. Then by induction hypothesis | V T ′ ( f i ) ∩ V T ′ ( g i ) | ≤ ( q − s − − q ( q − d − ( q − s − d − . (2) If one of f i and g i has degree d −
1. Let us call it g ′ i , then by Lemma 3 . | V T ′ ( f i ) ∩ V T ′ ( g i ) | ≤ | V T ′ ( g ′ i ) |≤ ( q − s − − ( q − d − ( q − s − d . (3) If deg f i = deg g i = d and d < s −
1, then by induction hypothesis | V T ′ ( f i ) ∩ V T ′ ( g i ) | ≤ ( q − s − − q ( q − d ( q − s − d − . (4) If deg f i = deg g i = d and d = s −
1, then f i and g i are square-free polynomialsof degree d in d variables so the leading monomial of f i and g i are equal. Weconstruct square-free polynomials f ′ i and g ′ i as follows. f ′ i := f i and g ′ i := f i − LC ( f i ) LC ( g i ) g i . Then, g ′ i is a square-free polynomial of degree d ′ ≤ d − s − V T ′ ( f i , g i ) = V T ′ ( f ′ i , g ′ i ). Then | V T ′ ( f i ) ∩ V T ′ ( g i ) | ≤ | V T ′ ( g ′ i ) |≤ ( q − s − − ( q − d ′ ( q − s − d ′ − ≤ ( q − s − − ( q − d − ( q − s − d . Therefore, in equation (3 .
2) we have if d < s − | V T ( f ) ∩ V T ( g ) | ≤ q − X i =1 | V T ′ ( f i ) ∪ V T ′ ( g i ) |≤ ( q − q − s − − q ( q − d ( q − s − d − ] ≤ ( q − s − q ( q − d ( q − s − d − . But if d = s − , (2) and (4), we get | V T ( f ) ∩ V T ( g ) | ≤ q − X i =1 | V T ′ ( f i ) ∪ V T ′ ( g i ) |≤ ( q − q − s − − ( q − s − ( q − ≤ ( q − s − ( q − s − ( q − ≤ ( q − s − q ( q − s − = ( q − s − q ( q − d ( q − s − d − . (cid:3) Extending Lemma 3 . . S requiresdealing with many cases and is tiresome. Also, these lemmas calculate the cardinality ofthe sets when all the polynomials are of degree d . In the remaining part of this section, wegive an upper bound on the number of zeroes in the affine torus T = ( F ∗ q ) s of square-freepolynomials of degree d . Then we obtain an upper bound on the number of zeroes in T ENERALIZED HAMMING WEIGHTS OF CODES 11 of square-free polynomials of degree at most d .Let s ≥
2, 1 ≤ d ≤ s be fixed and 1 ≤ r ≤ (cid:0) sd (cid:1) . Let f , f , · · · , f r ∈ KV ≤ d belinearly independent polynomials of degree d . We assume that their leading monomialsare distinct. To calculate the number of zeroes of f , f , · · · , f r in T we use the ideas of[6].We have I = I ( T ) is the vanishing ideal of T in S . The set { t q − i − i = 1 , · · · , s } is aGroebner basis of I . The ideal L := h LT ( I ) i is generated by the set { t q − i : i = 1 , · · · , s } . Let J := LT ( I ( V T ( f , f , · · · , f r ))) and for i = 1 , , · · · , r , let t a i = t a i, t a i, · · · t a i,s s := LT ( f i ). Consider the ideal A := h t q − , t q − , · · · , t q − s , t a , t a , · · · , t a r i . From Proposition2 . . | V T ( f , · · · , f r ) | = a HF J ( u ) ≤ a HF A ( u ) , for all sufficiently large u . Thus, our next goal is to calculate a HF A ( u ). Before that, weintroduce the following notations as in [6]. Definition 3.4.
Let k := s ( q − . (1) F := ( { , , · · · , q − } ) s and G := ( { , } ) s . (2) For b := ( b , b , · · · , b s ) ∈ F , define deg ( b ) = b + b · · · + b s . (3) For u ≤ k , define F u := { b ∈ F : deg ( b ) = u } and F ≤ u := { b ∈ F : deg ( b ) ≤ u } . (4) ( b , b , · · · , b s ) ≤ P ( c , c , · · · , c s ) if and only if b ≤ c , b ≤ c , · · · , b s ≤ c s . (5) For H ⊆ F , define shadow of H as ∇ F ( H ) := { a ∈ F : b ≤ P a for some b ∈ H } . Following the idea as in [6], we write A as A = A + A where A = h t q − , t q − , · · · , t q − s i and A = h t a , t a , · · · , t a s i . Then any monomial t b := t b t b · · · t b s s that doesn’t belong to A has b i ≤ q −
2, for all i , 1 ≤ i ≤ s . Now, if M A denotes the set of monomials thatdoes not belong to A , then M A is in bijection with the set F . From Proposition 2 . t b ∈ M A will belong to A if and only if t a j | t b for some j , 1 ≤ j ≤ r i.e. a j ≤ P b . Also, { a , a , · · · , a r } ⊆ G d . Thus, we have a HF A ( u ) = | F \∇ F ( a , · · · , a r ) | , where u ≥ k . Hence,(3.3) | V T ( f , · · · f r ) | ≤ max { | F \∇ F ( a , a , · · · , a r ) | : a , a , · · · , a r ∈ G d } . The following lemma gives a lower bound on |∇ F ( { a , a , · · · , a r } ) | for a , a , · · · , a r ∈ G d . Lemma 3.5.
Let s ≥ , ≤ d ≤ s and ≤ r ≤ (cid:0) sd (cid:1) . If d + r − < s then for any B = { a , a , · · · , a r } ⊆ G d with | B | = r , we have |∇ F ( B ) | ≥ ( q − d − ( q − s − d − r +1 [( q − r − . Proof.
We prove this lemma by induction on r . For r = 1, the lemma is clearly true. For r = 2, let B = { a, b } ⊆ G d where a = ( a , a , · · · , a s ) and b = ( b , b , · · · , b s ). Define forany v = ( v , v , · · · , v s ) ∈ F , supp v := { ≤ i ≤ s : v i = 0 } . Let A := supp a ∪ supp b and | A | =: e . Then |∇ F ( B ) | = s Y i =1 ( q − − a i ) + s Y i =1 ( q − − b i ) − s Y i =1 min ( q − − a i , q − − b i )= 2( q − d ( q − s − d − ( q − e ( q − s − e . To prove the lemma for r = 2, we have to show that2( q − d ( q − s − d − ( q − e ( q − s − e ≥ q ( q − d ( q − s − d − , which is equivalent to proving that(3.4) ( q − d +1 ( q − s − d − ≥ ( q − e ( q − s − e . Observe that e ≥ d + 1, so equation (3 .
4) holds.Assume that the lemma is true for r −
1. We prove it for r . For any B = { a , · · · , a r } ⊆ G d , ∇ F ( B ) = ∇ F ( { a , a , · · · , a r } ) = ∇ F ( { a , a , · · · , a r − } ) ∪ [ ∇ ( { a r } ) \∇ F ( { a , a , · · · , a r − } )] . Therefore(3.5) |∇ F ( B ) | = |∇ F ( { a , a , · · · , a r − } ) | + |∇ ( { a r } ) \ ∇ F ( { a , a , · · · , a r − } ) | . We arrange a , a , · · · , a r in graded lexicographic order. Without loss of generalityassume that a r ≺ · · · ≺ a ≺ a .Now, look at a , find a position p (1 ≤ p ≤ s ) such that any b ∈ F with b p = 0doesn’t belong to ∇ F ( { a } ) but belongs to ∇ F ( { a r } ). (It may or may not belong to ∇ F ( { a , · · · , a r − } ). Next, look at a and find a position p similarly ( Note that p maybe equal to p ). Keep on doing this upto a r − . In this way, we get p , p , · · · , p r − . Let e , e , · · · , e w denotes distinct elements from p , p , · · · , p r − . Then 1 ≤ w ≤ r −
1. Let1 ≤ v , v , · · · , v d ≤ s be the positions where a r is non-zero.Consider b = ( b , b , · · · , b s ) ∈ F with b e i = 0 for 1 ≤ i ≤ w and b v j = 1 for 1 ≤ j ≤ d. Then, any such b is contained in ∇ F ( { a r } ) but not in ∇ F ( { a , · · · , a r − } ). The cardinalityof set of all such b ’s is ( q − d ( q − s − w − d ≥ ( q − d ( q − s − d − r +1 .By induction hypothesis, we obtain from equation (3 . |∇ F ( B ) | ≥ ( q − d − ( q − s − d − r +2 [( q − r − −
1] + ( q − d ( q − s − d − r +1 = ( q − d − ( q − s − d − r +1 [( q − r − . ENERALIZED HAMMING WEIGHTS OF CODES 13 (cid:3)
Now, we state a lemma which will be required in our main result. The proof is similarto [1], Lemma 3 . Lemma 3.6.
Let L be a K -linear subspace of S = K [ t , · · · , t s ] of finite dimension andlet F ′ = { f , f , · · · , f r } be a subset of L\{ } . If f , f , · · · , f r are linearly independentover K , then there is a set G ′ = { g , g , · · · , g r } ⊂ L\{ } such that • KF ′ = KG ′ . • LM ( g ) , LM ( g ) , · · · , LM ( g r ) are distinct. • LM ( g i ) (cid:22) LM ( f i ) for all i . • g , g , · · · , g r are linearly independent over K . • V T ( F ′ ) = V T ( G ′ ) .Proof. We proceed by induction on r . The case r = 1 is clear. Let r = 2. For F ′ = { f , f } ⊂ L\{ } linearly independent set over K . If LM ( f ) = LM ( f ), then G ′ := F ′ works. Otherwise, define g := f and g := f − LC ( f ) LC ( f ) f . Then G ′ := { g , g } ⊂ L\{ } such that KF ′ = KG ′ , LM ( g ) = LM ( g ) and LM ( g i ) (cid:22) LM ( f i ) for i = 1 ,
2. Also, G ′ is linearly independent set over K and V T ( F ′ ) = V T ( G ′ ).Now, assume that r > LM ( f r ) (cid:22) · · · (cid:22) LM ( f ) (cid:22) LM ( f ). We have thefollowing two cases. • If LM ( f ) ≺ LM ( f ), then applying induction hypothesis to the set F ′′ = { f , f , · · · , f r } we obtain a set G ′′ = { g , g , · · · , g r } ⊂ L\{ } such that KF ′′ = KG ′′ , LM ( g ) , LM ( g ) · · · , LM ( g r ) are distinct, LM ( g i ) (cid:22) LM ( f i ) for i = 2 , , · · · , r and g , g , · · · , g r are linearly independent over K . Also V T ( F ′′ ) = V T ( G ′′ ). Define g := f and G ′ := G ′′ ∪ { g } . This implies KF ′ = KG ′ . Since LM ( g i ) (cid:22) LM ( f i ) ≺ LM ( f )for i = 2 , , · · · , r , the monomials LM ( g i ), 1 ≤ i ≤ r , are distinct. Also, G ′ islinearly independent over K and V T ( F ′ ) = V T ( f ) ∩ V T ( F ′′ ) = V T ( g ) ∩ V T ( G ′′ ) = V T ( G ′ ) . • If LM ( f ) = LM ( f ), assume that there exists l ≥ LM ( f ) = LM ( f i )for i ≤ l and LM ( f i ) ≺ LM ( f ) for i > l . Define h i = f − LC ( f ) LC ( f i ) f i for i = 2 , , · · · , l and h i = f i for i > l. Then LM ( h i ) ≺ LM ( f ) for i ≥ H = { h , h , · · · , h r } ⊂ L\{ } isa linearly independent set. By induction hypothesis for H , we obtain a set G ′′ = { g , g , · · · , g r } ⊂ L\{ } such that KH = KG ′′ , LM ( g i ) , i = 2 , , · · · , r are distinct and LM ( g i ) (cid:22) LM ( h i ) for i = 2 , , · · · , r . Also, G ′′ is linearly inde-pendent set over K and V T ( H ) = V T ( G ′′ ). Define g := f and G ′ := G ′′ ∪ { g } .We obtain G ′ ⊂ L\{ } such that LM ( g ) , LM ( g ) , · · · , LM ( g r ) are distinct. As LM ( g i ) (cid:22) LM ( h i ) ≺ LM ( f ) for i = 2 , , · · · , r , we have LM ( g i ) (cid:22) LM ( f i ) for i = 1 , , · · · , r . Also, G ′ is linearly independent set. Thus, as V T ( F ′ ) = V T ( f ) ∩ V T ( H ) = V T ( g ) ∩ V T ( G ′′ ) = V T ( G ′ ) . (cid:3) Thus, we get our main result.
Theorem 3.7.
Let s ≥ , ≤ d ≤ s and ≤ r ≤ (cid:0) sd (cid:1) . If d + r − < s , then for any f , f , · · · , f r ∈ KV ≤ d of linearly independent polynomials over K of degree d , we have | V T ( f , f , · · · , f r ) | ≤ ( q − s − ( q − d − ( q − s − d − r +1 [( q − r − . Proof.
Suppose leading monomials of f , f , · · · , f r are distinct, then the inequality followsfrom equation (3 .
3) and Lemma 3 .
5. If the leading monomials of f , f , · · · , f r are notall distinct, then by Lemma 3 .
6, we get g , g , · · · , g r with distinct leading monomials.If deg g i = d for all i . Then the theorem follows from equation (3 .
3) and Lemma 3 .
5. Ifatleast one g i has degree less than d , without loss of generality assume that deg g r = d ′ < d .Then, | V T ( f , f , · · · , f r ) | = | V T ( g , g , · · · , g r ) |≤ | V T ( g r ) |≤ ( q − s − ( q − d ′ ( q − s − d ′ ≤ ( q − s − ( q − d − ( q − s − d − r +1 [( q − r − . (cid:3) Now, Let f , f , · · · , f r ∈ KV ≤ d be linearly independent. Since the polynomials arelinearly independent, we can assume that their leading monomials are distinct. Followingthe procedure as before, we get(3.6) | V T ( f , · · · f r ) | ≤ max { | F \∇ F ( a , a , · · · , a r ) | : a , a , · · · , a r ∈ G ≤ d } . Repeating the procedure of Lemma 3 .
5, we have the following lemma.
Lemma 3.8.
Let s ≥ , ≤ d ≤ s and ≤ r ≤ dim KV ≤ d . If d + r − < s , then forany B = { a , a , · · · , a r } ⊆ G ≤ d with | B | = r , we have |∇ F ( B ) | ≥ ( q − d − ( q − s − d − r +1 [( q − r − . Proof.
We prove this lemma by induction on r . For r = 1, the inequality holds. For r = 2, let B = { a, b } ⊆ G ≤ d where a = ( a , a , · · · , a s ) and b = ( b , b , · · · , b s ). Let A := supp a , A := supp b and let | A | =: e , | A | =: e . Without loss of generality,suppose that e ≤ e ≤ d . Then |∇ F ( B ) | = s Y i =1 ( q − − a i ) + s Y i =1 ( q − − b i ) − s Y i =1 min ( q − − a i , q − − b i )= ( q − e ( q − s − e + ( q − e ( q − s − e − ( q − | A ∪ A | ( q − s −| A ∪ A | . ENERALIZED HAMMING WEIGHTS OF CODES 15
To prove the lemma for r = 2, we consider the following two cases as in [1]. • If e = d , then we have to show( q − d +1 ( q − s − d − ≥ ( q − | A ∪ A | ( q − s −| A ∪ A | , which we have already proved in previous lemma. • If e < d , then we the lemma holds true using the following inequalities( q − e ( q − s − e ≥ q ( q − d ( q − s − d − and ( q − e ( q − s − e ≥ ( q − | A ∪ A | ( q − s −| A ∪ A | .Assume that the lemma is true for r −
1. We prove for r . For any B = { a , a , · · · , a r } ⊆ G ≤ d , ∇ F ( B ) = ∇ F ( { a , a , · · · , a r } ) = ∇ F ( { a , a , · · · , a r − } ) ∪ [ ∇ ( { a r } ) \∇ F ( { a , a , · · · , a r − } )] . Therefore,(3.7) |∇ F ( B ) | = |∇ F ( { a , a , · · · , a r − } ) | + |∇ ( { a r } ) \ ∇ F ( { a , a , · · · , a r − } ) | . We arrange a , a , · · · , a r in graded lexicographic order. Without loss of generality,assume that a r ≺ · · · ≺ a ≺ a .Now, look at a , find a position p (1 ≤ p ≤ s ) such that any b ∈ F with b p = 0doesn’t belong to ∇ F ( { a } ) but belongs to ∇ F ( { a r } ). (It may or may not belong to ∇ F ( { a , · · · , a r − } ). Next, look at a and find a position p similarly ( Note that p maybe equal to p ). Keep on doing this upto a r − . In this way, we get p , p , · · · , p r − . Let e , e , · · · , e w denotes distinct elements from p , p , · · · , p r − . Then 1 ≤ w ≤ r −
1. Let1 ≤ v , v , · · · , v l ≤ s be the positions where a r is non-zero. Note that l ≤ d . If l = 0,then a r = (0 , · · · ,
0) and |∇ F ( B ) | = ( q − s ≥ ( q − d − ( q − s − d − r +1 [( q − r − l ≥ b = ( b , b , · · · , b s ) ∈ F with b e i = 0 for 1 ≤ i ≤ w and b v j = 1 for 1 ≤ j ≤ l. Then any such b is contained in ∇ F ( { a r } ) but not in ∇ F ( { a , · · · , a r − } ). The cardinalityof set of all such b ’s is ( q − l ( q − s − w − l ≥ ( q − d ( q − s − d − r +1 .By induction hypothesis, we obtain from equation (3 . |∇ F ( B ) | ≥ ( q − d − ( q − s − d − r +2 [( q − r − −
1] + ( q − d ( q − s − d − r +1 = ( q − d − ( q − s − d − r +1 [( q − r − . (cid:3) Thus, we get
Theorem 3.9.
For ≤ r ≤ dim KV ≤ d , if f , · · · , f r ∈ KV ≤ d are linearly independentand d + r − < s , then | V T ( f , · · · , f r ) | ≤ ( q − s − ( q − d − ( q − s − d − r +1 [( q − r − . Generalized Hamming weights of certain evaluation codes
Let s and d be integers such that s ≥ ≤ d ≤ s . In this section, we determinethe generalized Hamming weights of toric codes C d and C P d over hypersimplices, as definedin section 2 .
1. We also determine the generalized Hamming weights of square-free affineevaluation code C ≤ d , as defined in section 2 . Generalized Hamming weights of C d and C P d . For 1 ≤ r ≤ dim K KV d , the r -thgeneralized Hamming weight d r ( C P d ) of C P d is given by d r ( C P d ) := min { | T |\| V T ( H ) | : H := { f , f , · · · , f r } ⊆ KV d is linearly independent over K } . Similarly, we define d r ( C d ). (It follows from [1] and [4]).In this subsection, we find formulae for generalized Hamming weights of codes C d and C P d under certain cases. Theorem 4.1.
Let ≤ r ≤ (cid:0) sd (cid:1) . For d + r − < s , we have d r ( C P d ) = ( q − d − ( q − s − d − r [( q − r − . Proof.
For any f , f , · · · , f r ∈ KV d linearly independent over K , we have( q − | V T ( f , f , · · · , f r ) | = | V T ( f , f , · · · , f r ) | . So, from Theorem 3 . d + r − < d + r − < s , we get | V T ( f , f , · · · , f r ) | ≤ ( q − s − − ( q − d − ( q − s − d − r [( q − r − . Thus, d r ( C P d )) ≥ ( q − d − ( q − s − d − r [( q − r − . For the converse, consider the polynomials g , g , · · · , g r where g i := ( t − t )( t − t ) · · · ( t d − − t d − )( t d + i − − t d + i − ) , for 1 ≤ i ≤ r. Then g , g , · · · , g r ∈ KV d and are linearly independent over K . Let g := ( t − t )( t − ENERALIZED HAMMING WEIGHTS OF CODES 17 t ) · · · ( t d − − t d − ) and h i := ( t d + i − − t d + i − ) for 1 ≤ i ≤ r . Let T = ( F ∗ q ) d − . Then | V T ( g , g , · · · , g r ) | = | V T ( g ) ∩ V T ( g ) ∩ · · · ∩ V T ( g r ) | = | V T ( g ) ∪ ( V T ( h ) ∩ V T ( h ) ∩ · · · ∩ V T ( h r )) | = | V T ( g ) | + | V T ( h ) ∩ V T ( h ) ∩ · · · ∩ V T ( h r ) |− | V T ( g ) ∩ V T ( h ) ∩ V T ( h ) ∩ · · · ∩ V T ( h r ) | = ( q − s − d +2 | V T ( g ) | + ( q − s − r − ( q − s − d − r +2 | V T ( g ) | = ( q − s − r + ( q − s − d − r +2 [( q − r − | V T ( g ) | = ( q − s − r + ( q − s − d − r +2 [( q − r − " d − X i =1 ( − i − (cid:18) d − i (cid:19) ( q − d − − i = ( q − s − ( q − s − d − r +1 ( q − d − [( q − r − q − | V T ( g , g , · · · , g r ) | . Thus, d r ( C P d ) ≤ ( q − d − ( q − s − d − r [( q − r − . Hence, the result follows. (cid:3)
Corollary 4.2.
Let ≤ r ≤ (cid:0) sd (cid:1) . For d + r − < s , we have d r ( C d ) = ( q − d − ( q − s − d − r +1 [( q − r − , Now, consider the following definition.
Definition 4.3.
For a homogeneous polynomial f ∈ S of degree d , we define the polyno-mial f ∗ ( t , t , · · · , t s ) := t t · · · t s f ( t − , t − , · · · , t − s ) . Then, f ∗ ∈ S and is of degree s − d . Also, ( a , a , · · · , a s ) ∈ V T ( f ) if and only if( a − , a − , · · · , a − s ) ∈ V T ( f ∗ ). Theorem 4.4.
Let ≤ r ≤ (cid:0) sd (cid:1) . For s < d − r + 2 , we have d r ( C P d ) = ( q − s − d − ( q − d − r [( q − r − . Proof.
For f , f , · · · , f r ∈ KV d linearly independent polynomials, we have | V T ( f , f , · · · , f r ) | = | V T ( f ∗ , f ∗ , · · · , f ∗ r ) | . Now, f ∗ , f ∗ , · · · , f ∗ r ∈ KV s − d . Also, if v = s − d then 2 v + r − < s . Thus, by Theorem4 .
1, we have | V T ( f ∗ , f ∗ , · · · , f ∗ r ) | ≤ ( q − s − − ( q − v − ( q − s − v − r [( q − r − . Therefore, d r ( C P d )) ≥ ( q − s − d − ( q − d − r [( q − r − . For the converse, consider the polynomials g ′ , g ′ , · · · , g ′ r where g ′ := ( t − t )( t − t ) · · · ( t v − − t v − )( t v − − t v ) t v +1 t v +2 · · · t s ,g ′ := ( t − t )( t − t ) · · · ( t v − − t v − )( t v − t v +1 ) t v − t v +2 t v +3 · · · t s ,g ′ := ( t − t )( t − t ) · · · ( t v − − t v − )( t v +1 − t v +2 ) t v − t v t v +3 · · · t s , ... g ′ r := ( t − t )( t − t ) · · · ( t v − − t v − )( t v + r − − t v + r − ) t v − t v · · · t v + r − t v + r · · · t s . Then g ′ , g ′ , · · · g ′ r ∈ KV d and are linearly independent. Let g := ( t − t )( t − t ) · · · ( t v − − t v − ) and h i := ( t v + i − − t v + i − ), for 1 ≤ i ≤ r . Let T = ( F ∗ q ) v − . Then proceeding asin Theorem 4 .
1, we get | V T ( g ′ , g ′ , · · · , g ′ r ) | = | V T ( g ′ ) ∩ V T ( g ′ ) ∩ · · · ∩ V T ( g ′ r )) | = | V T ( g ) ∪ ( V T ( h ) ∩ V T ( h ) ∩ · · · ∩ V T ( h r )) | = ( q − s − v +2 | V T ( g ) | + ( q − s − r − ( q − s − v − r +2 | V T ( g ) | = ( q − s − r + ( q − s − v − r +2 [( q − r − " v − X i =1 ( − i − (cid:18) v − i (cid:19) ( q − v − − i = ( q − s − ( q − s − v − r +1 ( q − v − [( q − r − q − s − ( q − d − r +1 ( q − s − d − [( q − r − q − | V T ( g ′ , g ′ , · · · , g ′ r ) | This implies d r ( C P d ) ≤ ( q − s − d − ( q − d − r [( q − r − . This proves the result. (cid:3)
Corollary 4.5.
Let ≤ r ≤ (cid:0) sd (cid:1) . For s < d − r + 2 , we have d r ( C d ) = ( q − s − d − ( q − d − r +1 [( q − r − . If r = 1, Theorem 4 .
1, Theorem 4 .
4, Corollary 4 . . C d and C P d as in Theorem 2 .
2. If r = 2, these results determine thesecond generalized Hamming weight of C d and C P d for s > d and s < d . Similarly, if r = 3, we get the third generalized Hamming weight of these codes for s > d + 1 and s < d −
1. We give estimates on the second and third generalized Hamming weights of C d and C P d , in the rest of the cases.Note that, for q = 2 or s = d , dim F q C d = dim F q C P d = 1. Thus, the second generalizedHamming weight of these codes doesn’t make sense. So, we assume that q ≥ d < s when calculating the second generalized weight of these codes. Similarly, we assume that (cid:0) sd (cid:1) ≥ q ≥ ENERALIZED HAMMING WEIGHTS OF CODES 19
Proposition 4.6.
For s = 2 d , there exists f , f ∈ KV d linearly independent over K suchthat | V T ( f ) ∩ V T ( f ) | = ( q − s − − ( q − d − ( q − s − d . Therefore, d ( C P d ) ≤ ( q − d − ( q − s − d and d ( C d ) ≤ ( q − d − ( q − s − d +1 . Proof.
Consider the polynomials f ′′ := ( t − t )( t − t ) · · · ( t d − − t d − )( t d − − t d )and f ′′ := ( t − t )( t − t ) · · · ( t d − − t d − ) t d . Then f ′′ , f ′′ ∈ KV d are linearly independent over K . For 1 ≤ i ≤ d , let h i := ( t i − − t i ).Then | V T ( f ′′ ) ∩ V T ( f ′′ ) | = | ( V T ( h h · · · h d − ) ∪ V T ( h d )) ∩ ( V T ( h h · · · h d − ) ∪ V T ( t d )) | = | V T ( h h · · · h d − ) | = d − X i =1 ( − i − (cid:18) d − i (cid:19) ( q − s − i = ( q − s − ( q − s − d +1 ( q − d − = ( q − | V T ( f ′′ ) ∩ V T ( f ′′ ) | . Thus, for s = 2 d , d ( C P d ) ≤ ( q − d − ( q − s − d and d ( C d ) ≤ ( q − d − ( q − s − d +1 . (cid:3) Proposition 4.7.
Let s = 2 d + 1 and (cid:0) sd (cid:1) ≥ . There exists f , f , f ∈ KV d linearlyindependent over K such that | V T ( f ) ∩ V T ( f ) ∩ V T ( f ) | = ( q − s − − ( q − d − ( q − s − d . Therefore, d ( C P d ) ≤ ( q − d − ( q − s − d and d ( C d ) ≤ ( q − d − ( q − s − d +1 . Proof.
Consider the following polynomials f := ( t − t )( t − t ) · · · ( t d − − t d − ) t d − ,f := ( t − t )( t − t ) · · · ( t d − − t d − ) t d , and f := ( t − t )( t − t ) · · · ( t d − − t d − ) t d +1 . Let g := ( t − t )( t − t ) · · · ( t d − − t d − ). Then, | V T ( f ) ∩ V T ( f ) ∩ V T ( f ) | = | V T ( g ) | = d − X i =1 ( − i − (cid:18) d − i (cid:19) ( q − s − i = ( q − s − ( q − d − ( q − s − d +1 = ( q − | V T ( f ) ∩ V T ( f ) ∩ V T ( f ) | . This implies d ( C P d ) ≤ ( q − d − ( q − s − d and d ( C d ) ≤ ( q − d − ( q − s − d +1 . (cid:3) Proposition 4.8.
Let s = 2 d and (cid:0) sd (cid:1) ≥ . There exists f , f , f ∈ KV d linearly inde-pendent over K such that | V T ( f ) ∩ V T ( f ) ∩ V T ( f ) | = ( q − s − − ( q − d − ( q − s − d − [ q ( q − − . Therefore, d ( C P d ) ≤ ( q − d − ( q − d − [ q ( q − − and d ( C d ) ≤ ( q − d − ( q − d − [ q ( q − − . Proof.
Let T := ( F ∗ q ) d − and consider the polynomials f := ( t − t )( t − t ) · · · ( t d − − t d − )( t d − − t d − )( t d − − t d ) ,f := ( t − t )( t − t ) · · · ( t d − − t d − )( t d − − t d − )( t d − − t d ) , and f := ( t − t )( t − t ) · · · ( t d − − t d − )( t d − − t d − ) t d . Let g := ( t − t )( t − t ) · · · ( t d − − t d − ). Then, | V T ( f ) ∩ V T ( f ) ∩ V T ( f ) | = | V T ( g ) ∪ V T ( t d − − t d − , t d − − t d − ) ∪ V T ( t d − − t d , t d − − t d − ) | = 2( q − s − − ( q − s − + ( q − s − d +1 [( q − − q −
1) + 1] | V T ( g ) | = 2( q − s − − ( q − s − + [( q − s − d +4 − q − s − d +2 + ( q − s − d +1 ] " − d − X i =1 ( − i (cid:18) d − i (cid:19) ( q − d − − i = ( q − s − ( q − d − ( q − d − [ q ( q − − q − | V T ( f ) ∩ V T ( f ) ∩ V T ( f ) | . Therefore, d ( C P d ) ≤ ( q − d − ( q − d − [ q ( q − − d ( C d ) ≤ ( q − d − ( q − d − [ q ( q − − . (cid:3) ENERALIZED HAMMING WEIGHTS OF CODES 21
Proposition 4.9.
Let s = 2 d − and (cid:0) sd (cid:1) ≥ . There exists f , f , f ∈ KV d linearlyindependent over K such that | V T ( f ) ∩ V T ( f ) ∩ V T ( f ) | = ( q − s − − ( q − d − ( q − s − d +1 . Therefore, d ( C P d ) ≤ ( q − d − ( q − s − d +1 and d ( C d ) ≤ ( q − d − ( q − s − d +2 . Proof.
Consider the following polynomials f := ( t − t )( t − t ) · · · ( t d − − t d − ) t d − t d − ,f := ( t − t )( t − t ) · · · ( t d − − t d − ) t d − t d − , and f := ( t − t )( t − t ) · · · ( t d − − t d − ) t d − t d − . Let g := ( t − t )( t − t ) · · · ( t d − − t d − ). Then, | V T ( f ) ∩ V T ( f ) ∩ V T ( f ) | = | V T ( g ) | = d − X i =1 ( − i − (cid:18) d − i (cid:19) ( q − s − i = ( q − s − ( q − d − ( q − s − d +2 = ( q − | V T ( f ) ∩ V T ( f ) ∩ V T ( f ) | . Therefore, d ( C P d ) ≤ ( q − d − ( q − s − d +1 and d ( C d ) ≤ ( q − d − ( q − s − d +2 . (cid:3) Generalized Hamming weights of square-free affine evaluation code.
For1 ≤ r ≤ dim K KV ≤ d , the r -th generalized Hamming weight of C ≤ d is given by d r ( C ≤ d ) := min { | T \ V T ( H ) | : H := { f , f , · · · , f r } ⊆ KV ≤ d is linearly independent over K } . In this subsection, we determine the generalized Hamming weights of C ≤ d , partially. Theorem 4.10.
Let ≤ r ≤ dim K KV ≤ d . For d + r − < s , we have d r ( C ≤ d ) = ( q − d − ( q − s − d − r +1 [( q − r − . Proof.
From Theorem 3 .
9, we have d r ( C ≤ d ) ≥ ( q − d − ( q − s − d − r +1 [( q − r − . To prove the converse, consider the following polynomials f ′ = ( t − t − · · · ( t d − − t d − ,f ′ = ( t − t − · · · ( t d − − t d +1 − , ... f ′ r = ( t − t − · · · ( t d − − t d + r − − . Then, f ′ , f ′ , · · · , f ′ r ∈ KV ≤ d are linearly independent over K . Let g = ( t − t − · · · ( t d − − T = ( F ∗ q ) d − . Then, | V T ( f ′ , f ′ , · · · , f ′ r ) | = | V T ( g ) ∪ V T ( t d − , t d +1 − , · · · , t d + r − − | = ( q − s − d +1 | V T ( g ) | + ( q − s − r − ( q − s − d − r +1 | V T ( g ) | = ( q − s − r + ( q − s − d − r +1 [( q − r − q − d − − ( q − d − ]= ( q − s − ( q − d − ( q − s − d − r +1 [( q − r − . Thus, d r ( C ≤ d ) ≤ ( q − d − ( q − s − d − r +1 [( q − r − (cid:3) When r = 2, we get d ( C ≤ d ) = q ( q − d ( q − s − d − for d < s , which gives us result ofTheorem 2 . d < s . 5. Dual code
In this section, we determine the dual code of the toric codes over hypersimplices, usingthe ideas of [7] and [8].Consider the set(5.1) ∆ := { ( a , a , · · · , a s ) | a i ∈ { , } , s X i =1 a i = d } . For ( b , b , · · · , b s ) ∈ ( { , , · · · , q − } ) s , define ( b b , b b , · · · , b b s ) ∈ ( { , , · · · , q − } ) s as b b i = ( b i = 0 ,q − − b i if b i = 0 . Let G := { b b : b ∈ ∆ } and ∆ ′ := ( { , , · · · , q − } ) s \G . Define E ∆ ′ := span F q { t a t a · · · t a s s : ( a , a , · · · , a s ) ∈ ∆ ′ } . Replacing KV d by S in equation (2 . ev T : S → F mq f ( f ( P ) , f ( P ) , · · · , f ( P m )) , where T = { P , P , · · · , P m } is the affine torus in A s . Then, C ∆ ′ := ev T ( E ∆ ′ ) is alinear code over F q of length m on the affine torus T . Similarly, we have the code C ∆ = ev T ( E ∆ ) = C d . From the definition, it is clear that the map ev T | E ∆ ′ is injective. Therefore,we have Lemma 5.1. dim F q ( C ∆ ′ ) = ( q − s − (cid:0) sd (cid:1) = m − dim F q ( C d ) . ENERALIZED HAMMING WEIGHTS OF CODES 23
Lemma 5.2.
For a ∈ ∆ and b ∈ ∆ ′ , we have ev T ( t a ) .ev T ( t b ) = 0 , where we have t a := t a t a · · · t a s s for a = ( a , a , · · · , a s ) .Proof. Fix a primitive element θ of F q i.e. F ∗ q = h θ i . For a, b ∈ ( { , , · · · , q − } ) s , wehave ev T ( t a ) .ev T ( t b ) = s Y j =1 q − X i =0 ( θ i ) a j + b j . Now, if for some j , a j = b j = 0 or a j = q − − b j , then q − X i =0 ( θ i ) a j + b j = ( q − = 0 . But if for some j , a j + b j q − q − X i =0 ( θ i ) a j + b j = ( θ a j + b j ) q − − θ a j + b j − . Thus, ev T ( t a ) .ev T ( t b ) = 0 if and only if for each j , 1 ≤ j ≤ s , a j = b j = 0 or a j = q − − b j ,i.e. a ∈ ∆ and b ∆ ′ . Hence proved. (cid:3) Theorem 5.3.
The dual of the code C d is the code C ∆ ′ with respect to Euclidean scalarproduct i.e. C ∆ ′ = C ⊥ d . Proof.
Let f ∈ E ∆ ′ . Then f can be written as f = P b ∈ ∆ ′ α b t b where α b ∈ F q . For any g ∈ E ∆ , g = P a ∈ ∆ β a t a , β a ∈ F q , we have ev T ( g ) .ev T ( f ) = ev T X a ∈ ∆ β a t a ! .ev T X b ∈ ∆ ′ α b t b ! = X a ∈ ∆ X b ∈ ∆ ′ α b β a ev T ( t a ) .ev T ( t b ) = 0 , by Lemma 5 .
2. This implies C ∆ ′ ⊆ C ⊥ d . From Lemma 5 .
1, we have dim F q ( C ∆ ′ ) = dim F q ( C ⊥ d ). Hence the result. (cid:3) The codes C d and C ∆ ′ are J -affine variety codes with J = { , , · · · , s } , as studied in[7],[8],[9], etc. By using Corollary 2 from [7], we can obtain stabilizer codes.5.1. Dual of C P d . We have T = { } × ( F ∗ q ) s − . With ∆ as in equation (5 . H := { ( a , a , · · · , a s ) : ( a , a , · · · , a s ) ∈ ∆ } . For ( c , c , · · · , c s ) ∈ ( { , , · · · , q − } ) s − , define ( b c , b c , · · · , b c s ) ∈ ( { , , · · · , q − } ) s − as b c i = ( c i = 0 ,q − − c i if c i = 0 . Let H := { ( b b , b b , · · · , b b s ) : ( b , b , · · · , b s ) ∈ H } . Let U := ( { , , · · · , q − } ) s − \H .Define E U := span F q { t a t a · · · t a s s | ( a , a , · · · , a s ) ∈ U } .Let T ′ := ( F ∗ q ) s − . Then | T ′ | = ¯ m . Let T ′ = { R , R , · · · , R ¯ m } such that Q i = (1 , R i ),1 ≤ i ≤ ¯ m , where Q i ∈ { } × ( F ∗ q ) s − , 1 ≤ i ≤ ¯ m , as defined in section 2. Define a map ev T ′ : S → F ¯ mq f ( f ( R ) , f ( R ) , · · · , f ( R ¯ m )) . Define C U := ev T ′ ( E U ). Then C U is a linear code over F q of length ¯ m .Note that C P d = { ( f ( Q ) , · · · , f ( Q ¯ m )) : f ∈ KV d } = { ( f (1 , R ) , · · · , f (1 , R ¯ m )) : f ∈ KV d } = { ( g ( R ) , · · · , g ( R ¯ m )) : g ∈ E H } = ev T ′ ( E H ) , where E H is the F q -vector space generated by the set { t a := t a t a · · · t a s s : ( a , a , · · · , a s ) ∈H } . Then, dim F q C U = ( q − s − − (cid:0) sd (cid:1) = ¯ m − dim F q ( C P d ).Following Lemma 5 .
2, we get
Lemma 5.4.
For a ∈ H and b ∈ U , we have ev T ′ ( t a ) .ev T ′ ( t b ) = 0 . We have the final result.
Theorem 5.5.
The dual code of C P d is the code C U with respect to Euclidean scalar producti.e ( C P d ) ⊥ = C U . Concluding remarks
In this note, we have determined the generalized Hamming weights of toric codes overhypersimplices. The generalized Hamming weights of square-free affine evaluation codesare also calculated, under certain conditions. Furthermore, we have determined the dualof the toric codes with respect to the Euclidean scalar product. It will be interesting tocalculate the remaining generalized Hamming weights of these codes.
ENERALIZED HAMMING WEIGHTS OF CODES 25 Acknowledgements
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