The minimum distance of parameterized codes on projective tori
aa r X i v : . [ m a t h . A C ] J u l THE MINIMUM DISTANCE OF PARAMETERIZED CODES ONPROJECTIVE TORI
ELISEO SARMIENTO, MARIA VAZ PINTO, AND RAFAEL H. VILLARREAL
Abstract.
Let X be a subset of a projective space, over a finite field K , which is parameterizedby the monomials arising from the edges of a clutter. Let I ( X ) be the vanishing ideal of X . Itis shown that I ( X ) is a complete intersection if and only if X is a projective torus. In this casewe determine the minimum distance of any parameterized linear code arising from X . Introduction
Let K = F q be a finite field with q elements and let y v , . . . , y v s be a finite set of monomials.As usual if v i = ( v i , . . . , v in ) ∈ N n , then we set y v i = y v i · · · y v in n , i = 1 , . . . , s, where y , . . . , y n are the indeterminates of a ring of polynomials with coefficients in K . Considerthe following set parameterized by these monomials X := { [( x v · · · x v n n , . . . , x v s · · · x v sn n )] ∈ P s − | x i ∈ K ∗ for all i } , where K ∗ = K \ { } and P s − is a projective space over the field K . Following [20] we call X an algebraic toric set parameterized by y v , . . . , y v s . The set X is a multiplicative group undercomponentwise multiplication.Let S = K [ t , . . . , t s ] = ⊕ ∞ d =0 S d be a polynomial ring over the field K with the standardgrading, let [ P ] , . . . , [ P m ] be the points of X , and let f ( t , . . . , t s ) = t d . The evaluation map (1.1) ev d : S d = K [ t , . . . , t s ] d → K | X | , f (cid:18) f ( P ) f ( P ) , . . . , f ( P m ) f ( P m ) (cid:19) defines a linear map of K -vector spaces. This map is well defined, i.e., it is independent of thechoice of representatives P , . . . , P m . The image of ev d , denoted by C X ( d ), defines a linear code .Following [17] we call C X ( d ) a parameterized code of order d . As usual by a linear code we meana linear subspace of K | X | .The definition of C X ( d ) can be extended to any finite subset X ⊂ P s − of a projective spaceover a field K . Indeed if we choose a degree d ≥
1, for each i there is f i ∈ S d such that f i ( P i ) = 0and we can define C X ( d ) as the image of the evaluation map given byev d : S d = K [ t , . . . , t s ] d → K | X | , f (cid:18) f ( P ) f ( P ) , . . . , f ( P m ) f m ( P m ) (cid:19) . In this generality—the resulting linear code— C X ( d ) is called an evaluation code associated to X [9]. It is also called a projective Reed-Muller code over the set X [5, 12]. Some families of Mathematics Subject Classification.
Primary 13P25; Secondary 14G50, 14G15, 11T71, 94B27, 94B05.The first author was partially supported by CONACyT. The second author is a member of the Center for Math-ematical Analysis, Geometry and Dynamical Systems. The third author was partially supported by CONACyTgrant 49251-F and SNI. evaluation codes—including several variations of Reed-Muller codes—have been studied exten-sively using commutative algebra methods (e.g., Hilbert functions, resolutions, Gr¨obner bases),see [4, 5, 9, 10, 12, 17, 18, 19, 23]. In this paper we use these methods to study parameterizedcodes over finite fields. There are some other papers that have studied evaluation codes fromthe commutative algebra perspective [3, 14, 26].The dimension and the length of C X ( d ) are given by dim K C X ( d ) and | X | respectively. Thedimension and the length are two of the basic parameters of a linear code. A third basicparameter is the minimum distance which is given by δ d = min {k v k : 0 = v ∈ C X ( d ) } , where k v k is the number of non-zero entries of v . The basic parameters of C X ( d ) are related bythe Singleton bound for the minimum distance: δ d ≤ | X | − dim K C X ( d ) + 1 . The parameters of evaluation codes over finite fields have been computed in a number of cases.If X = P s − , the parameters of C X ( d ) are described in [23, Theorem 1]. If X is the image ofthe affine space A s − under the map A s − → P s − , x [(1 , x )], the parameters of C X ( d ) aredescribed in [4, Theorem 2.6.2]. Lower bounds for the minimum distance of evaluation codeshave been shown when X is any complete intersection reduced set of points in a projective space[3, 9, 14], and when X is a reduced Gorenstein set of points [26]. Upper bounds for the minimumdistance of certain parameterized codes are given in [17, 21]. In this paper we examine the casewhen X is an algebraic toric set parameterized by y , . . . , y s .The contents of this paper are as follows. In Section 2 we introduce the preliminaries andexplain the well known connection—via Hilbert functions—between the invariants of the van-ishing ideal of X and the parameters of C X ( d ), all the results of this section are well known. InSection 3 we recall a classical and well known upper bound for the number of roots of a non-zeropolynomial in S (see Lemmas 3.1 and 3.2). Then, we show upper bounds for the number ofroots, over an affine torus, for a certain family of polynomials in S (see Theorem 3.4). The maintheorem of Section 3 is a formula for the minimum distance of C X ( d ), where X = { [( x , . . . , x s )] ∈ P s − | x i ∈ K ∗ for all i } is a projective torus in P s − (see Theorem 3.5). Evaluation codes associated to a projective torusare called generalized projective Reed-Solomon codes [11]. If X is a projective torus in P or P ,we recover some formulas of [11, 17] for the minimum distance of C X ( d ) (see Proposition 3.6).Let X be an algebraic toric set parameterized by y v , . . . , y v s . The vanishing ideal of X ,denoted by I ( X ), is the ideal of S generated by the homogeneous polynomials of S that vanishon X . The ideal I ( X ) is called a complete intersection if it can be generated by s − S . In what follows we assume that v , . . . , v s are the characteristic vectors ofthe edges of a clutter (a special sort of hypergraph, see Definition 4.1). In Section 4 we areable to classify when I ( X ) is a complete intersection (see Theorem 4.4 and Corollary 4.5). Themain algebraic fact about I ( X ) that we need for this classification is a remarkable result of [17]showing that I ( X ) is a binomial ideal.The complete intersection property of I ( X ) has also been studied in [21], but from a linearalgebra perspective. Let φ : Z n /L → Z n /L be the multiplication map φ ( a ) = ( q − a , where L is the subgroup generated by { v i − v } si =2 . In [21] it is shown that if the clutter is uniform, i.e.,all its edges have the same cardinality, and q ≥
3, then I ( X ) is a complete intersection if andonly if v , . . . , v s are linearly independent and the map φ is injective. HE MINIMUM DISTANCE OF PARAMETERIZED CODES ON PROJECTIVE TORI 3
We show an optimal upper bound for the regularity of I ( X ) in terms of the regularity ofa complete intersection (see Proposition 4.6). This shows that the complete intersection I ( X )from clutters have the largest possible regularity.The ideal I ( X ) is studied in [21] from the viewpoint of computational commutative algebra.The degree-complexity and the reduced Gr¨obner basis of I ( X ), with respect to the reverselexicographical order, is examined in [21, Theorem 4.1].For all unexplained terminology and additional information we refer to [7] (for the theory ofbinomial ideals), [1, 24] (for the theory of polynomial ideals and Hilbert functions), [16, 25, 27](for the theory of error-correcting codes and algebraic geometric codes), and [17] (for the theoryof parameterized codes).2. Preliminaries: Hilbert functions and the basic parameters of codes
We continue to use the notation and definitions used in the introduction. In this section weintroduce the basic algebraic invariants of
S/I ( X ), via Hilbert functions, and we recall their wellknown connection with the basic parameters of parameterized linear codes. Then, we presentsome of the results that will be needed later.Recall that the projective space of dimension s − K , denoted by P s − , is the quotientspace ( K s \ { } ) / ∼ where two points α , β in K s \ { } are equivalent if α = λβ for some λ ∈ K ∗ . We denotethe equivalence class of α by [ α ]. Let X ⊂ P s − be an algebraic toric set parameterized by y v , . . . , y v s and let C X ( d ) be a parameterized code of order d . The kernel of the evaluation mapev d , defined in Eq. (1.1), is precisely I ( X ) d the degree d piece of I ( X ). Therefore there is anisomorphism of K -vector spaces S d /I ( X ) d ≃ C X ( d ) . It is well known that two of the basic parameters of C X ( d ) can be expressed using Hilbertfunctions of standard graded algebras [5, 12, 17, 23], as we now explain. Recall that the Hilbertfunction of S/I ( X ) is given by H X ( d ) := dim K ( S/I ( X )) d = dim K S d /I ( X ) d = dim K C X ( d ) . The unique polynomial h X ( t ) = P k − i =0 c i t i ∈ Z [ t ] of degree k − S/I ( X )) − h X ( d ) = H X ( d ) for d ≫ Hilbert polynomial of S/I ( X ). The integer c k − ( k − S/I ( X )), is called the degree or multiplicity of S/I ( X ). In our situation h X ( t ) isa non-zero constant because S/I ( X ) has dimension 1. Furthermore h X ( d ) = | X | for d ≥ | X | − | X | is the degree of S/I ( X ). Thus, H X ( d ) and deg( S/I ( X ))are the dimension and the length of C X ( d ) respectively.There are algebraic methods, based on elimination theory and Gr¨obner bases, to computethe dimension and the length of C X ( d ) [17]. This is one of the reasons that make some of thebasic parameters of parameterized codes more tractable. However, in general, the problem ofcomputing the minimum distance of a linear code is difficult because it is NP-hard [28].The index of regularity of S/I ( X ), denoted by reg( S/I ( X )), is the least integer p ≥ h X ( d ) = H X ( d ) for d ≥ p . The degree and the regularity index can be read off the Hilbertseries as we now explain. The Hilbert series of S/I ( X ) can be written as F X ( t ) := ∞ X i =0 H X ( i ) t i = ∞ X i =0 dim K ( S/I ( X )) i t i = h + h t + · · · + h r t r − t , ELISEO SARMIENTO, MARIA VAZ PINTO, AND RAFAEL H. VILLARREAL where h , . . . , h r are positive integers. Indeed h i = dim K ( S/ ( I ( X ) , t s )) i for 0 ≤ i ≤ r anddim K ( S/ ( I ( X ) , t s )) i = 0 for i > r . This follows from the fact that I ( X ) is a Cohen-Macaulaylattice ideal of height s − { t s } is a regular system of parametersfor S/I ( X ) (see [24]). The number r is the regularity index of S/I ( X ) and h + · · · + h r is thedegree of S/I ( X ) (see [29, Corollary 4.1.12]). In our situation, reg( S/I ( X )) is the Castelnuovo-Mumford regularity of S/I ( X ) [6]. We will refer to reg( S/I ( X )) as the regularity of S/I ( X ).For convenience we recall the following result on complete intersections. Proposition 2.1. [11, Theorem 1, Lemma 1] If T = { [( x , . . . , x s )] ∈ P s − | x i ∈ K ∗ for all i } is a projective torus in P s − , then (a) I ( T ) = ( { t q − i − t q − } si =2 ) . (b) F T ( t ) = (1 − t q − ) s − / (1 − t ) s . (c) reg( S/I ( T )) = ( s − q − and deg( S/I ( T )) = ( q − s − . When I ( X ) is a complete intersection, there is a general formula for the dimension of anyprojective Reed-Muller code arising from X [5]. For a projective torus one can easily find aformula for the dimension as shown below. Corollary 2.2. [5] If T is a projective torus in P s − , then the length of C T ( d ) is ( q − s − andits dimension is given by dim K C T ( d ) = j dq − k X j =0 ( − j (cid:18) s − j (cid:19)(cid:18) s − d − j ( q − s − (cid:19) . Proof.
According to Proposition 2.1, the length of C T ( d ) is ( q − s − and the Hilbert series ofthe graded algebra S/I ( T ) is given by F T ( t ) = ∞ X d =0 H T ( d ) t d = (1 − t q − ) s − (1 − t ) s = s − X j =0 ( − j (cid:18) s − j (cid:19) t j ( q − " ∞ X i =0 (cid:18) s − is − (cid:19) t i . Hence, comparing the coefficients of t d , we get H T ( d ) = X i + j ( q − d ( − j (cid:18) s − j (cid:19)(cid:18) s − is − (cid:19) . Thus making i = d − j ( q −
1) we obtain the required expression for dim K C T ( d ). (cid:3) In Section 3 we compute the minimum distance of C T ( d ), which was an important piece ofinformation—from the viewpoint of coding theory—missing in the literature.3. Minimum distance of parameterized codes
We continue to use the notation and definitions used in the introduction. In this section wedetermine the minimum distance of C X ( d ) when X is a projective torus in P s − .We begin with a well known and classical general upper bound. Lemma 3.1. [22, Lemma 3A, p. 147]
Let = G = G ( t , . . . , t s ) ∈ S be a polynomial of totaldegree d . Then the number N of zeros of G in F sq satisfies N ≤ dq s − . If G is homogeneous, then the number of its non-trivial zeros is at most d ( q s − − . HE MINIMUM DISTANCE OF PARAMETERIZED CODES ON PROJECTIVE TORI 5
The proof of this lemma, given in the book of W. M. Schmidt [22], can be easily adapted toobtain the following auxiliary result.
Lemma 3.2.
Let = G = G ( t , . . . , t s ) ∈ S be a polynomial of total degree d . If Z G := { x ∈ ( K ∗ ) s | G ( x ) = 0 } , then | Z G | ≤ d ( q − s − . Lemma 3.3.
Let d, d ′ , s be positive integers such that d = k ( q −
2) + ℓ and d ′ = k ′ ( q −
2) + ℓ ′ for some integers k, k ′ , ℓ, ℓ ′ satisfying that k, k ′ ≥ , ≤ ℓ ≤ q − and ≤ ℓ ′ ≤ q − . If d ′ ≤ d and k ≤ s − , then k ′ ≤ k and − ( q − s − k ′ + ℓ ′ ( q − s − k ′ − ≤ − ( q − s − k + ℓ ( q − s − k − . Proof.
It is not hard to see that k ′ ≤ k . It suffices to prove the equivalent inequality: q − − ℓ ≤ ( q − k − k ′ ( q − − ℓ ′ ) . If k = k ′ , then ℓ ≥ ℓ ′ and the inequality holds. If k ≥ k ′ + 1, then q − − ℓ ≤ q − ≤ ( q − q − − ℓ ′ ) ≤ ( q − k − k ′ ( q − − ℓ ′ ) , as required. (cid:3) Let T ∗ = ( K ∗ ) s be an affine torus . For G = G ( t , . . . , t s ) ∈ S , we denote the set of zeros of G in T ∗ by Z G . Theorem 3.4.
Let G = G ( t , . . . , t s ) ∈ S be a polynomial of total degree d ≥ such that deg t i ( G ) ≤ q − for i = 1 , . . . , s . If d = k ( q −
2) + ℓ with ≤ ℓ ≤ q − and ≤ k ≤ s − , then | Z G | ≤ ( q − s − k − (( q − k +1 − ( q −
1) + ℓ ) . Proof.
By induction on s . If s = 1, then k = 0 and d = ℓ . Then | Z G | ≤ ℓ because a non-zeropolynomial in one variable of degree d has at most d roots. Assume s ≥
2. By Lemma 3.2 wemay also assume that k ≥
1. There are r ≥ β , . . . , β r in K ∗ and G ′ ∈ S such that G = ( t − β ) a · · · ( t − β r ) a r G ′ , a i ≥ i, and G ′ ( β, t , . . . , t s ) = 0 for any β ∈ K ∗ . Notice that r ≤ P i a i ≤ q − G in t is at most q −
2. We can write K ∗ = { β , . . . , β q − } . Let d ′ i be the degree of G ′ ( β i , t , . . . , t s )and let d ′ = max { d ′ i | r + 1 ≤ i ≤ q − } . If d ′ = 0, then | Z G | = r ( q − s − and consequently r ( q − s − ≤ ( q − q − s − ≤ ( q − s − k − (( q − k +1 − ( q −
1) + ℓ ) . The second inequality uses that k ≥
1. Thus we may assume that d ′ > β r +1 , . . . , β m are the elements β i of { β r +1 , . . . , β q − } such that G ′ ( β i , t , . . . , t s ) has positivedegree. Notice that d = P i a i + deg( G ′ ) ≥ r + d ′ . The polynomial H := ( t − β ) a · · · ( t − β r ) a r has exactly r ( q − s − roots in ( K ∗ ) s . Hence counting the roots of G ′ that are not in Z H weobtain:(3.1) | Z G | ≤ r ( q − s − + m X i = r +1 | Z G ′ ( β i ,t ,...,t s ) | . For each r + 1 ≤ i ≤ m , we can write d ′ i = k ′ i ( q −
2) + ℓ ′ i , with 1 ≤ ℓ ′ i ≤ q −
2. The proof will bedivided in three cases.
ELISEO SARMIENTO, MARIA VAZ PINTO, AND RAFAEL H. VILLARREAL
Case (I): Assume ℓ > r and k = s −
1. By [2, Theorem 1.2], the non-zero polynomial G ′ ( β i , t , . . . , t s ) cannot be the zero-function on ( K ∗ ) s − for any i because its degree in each ofthe variables t , . . . , t s is at most q −
2. A direct argument to show that G ′ ( β i , t , . . . , t s ) cannotbe the zero-function on ( K ∗ ) s − is to notice that if this non-homogeneous polynomial vanisheson ( K ∗ ) s − , then it must be a polynomial combination of t q − − , . . . , t q − s −
1, a contradiction.Thus, by Eq. (3.1), we get | Z G | ≤ r ( q − s − + ( q − − r )(( q − s − − ≤ ( q − s − ( q −
1) + ℓ. Case (II): Assume ℓ > r and k ≤ s −
2. Then d − r = k ( q −
2) + ( ℓ − r ) with 1 ≤ ℓ − r ≤ q − d ′ i ≤ d − r for i = r + 1 , . . . , m , by Lemma 3.3, we get k ′ i ≤ k for r + 1 ≤ i ≤ m . Then byinduction hypothesis, using Eq. (3.1) and Lemma 3.3, we obtain: | Z G | ≤ r ( q − s − + m X i = r +1 | Z G ′ ( β i ,t ,...,t s ) |≤ r ( q − s − + m X i = r +1 h ( q − ( s − − k ′ i − (( q − k ′ i +1 − ( q −
1) + ℓ ′ i ) i ≤ r ( q − s − + ( q − − r ) h ( q − ( s − − k − (( q − k +1 − ( q −
1) + ( ℓ − r )) i ≤ ( q − s − k − (( q − k +1 − ( q −
1) + ℓ ) . Case (III): Assume ℓ ≤ r . Then we can write d − r = k ( q −
2) + ℓ with k = k − ℓ = q − ℓ − r . Notice that 0 ≤ k ≤ s − ≤ ℓ ≤ q − k ≥ r ≤ q − k ≤ s −
1. Since d ′ i ≤ d − r for i > r , by Lemma 3.3, we get k ′ i ≤ k for i = r + 1 , . . . , m . Thenby induction hypothesis, using Eq. (3.1) and Lemma 3.3, we obtain: | Z G | ≤ r ( q − s − + m X i = r +1 | Z G ′ ( β i ,t ,...,t s ) |≤ r ( q − s − + m X i = r +1 h ( q − ( s − − k ′ i − (( q − k ′ i +1 − ( q −
1) + ℓ ′ i ) i ≤ r ( q − s − + ( q − − r ) h ( q − ( s − − k − (( q − k +1 − ( q −
1) + ℓ ) i = r ( q − s − + ( q − − r ) h ( q − s − k − (( q − k − ( q −
1) + ( q − ℓ − r )) i ≤ ( q − s − k − (( q − k +1 − ( q −
1) + ℓ ) . The last inequality uses that r ≤ q −
2. This completes the proof of the result. (cid:3)
We come to the main result of this section.
Theorem 3.5. If X = { [( x , . . . , x s )] ∈ P s − | x i ∈ K ∗ for all i } is a projective torus in P s − and d ≥ , then the minimum distance of C X ( d ) is given by δ d = (cid:26) ( q − s − ( k +2) ( q − − ℓ ) if d ≤ ( q − s − − , if d ≥ ( q − s − , where k and ℓ are the unique integers such that k ≥ , ≤ ℓ ≤ q − and d = k ( q −
2) + ℓ .Proof. First we consider the case 1 ≤ d ≤ ( q − s − −
1. Then, in this case, we have that k ≤ s −
2. Let ≺ be the graded reverse lexicographical order on the monomials of S . In this HE MINIMUM DISTANCE OF PARAMETERIZED CODES ON PROJECTIVE TORI 7 order t ≻ · · · ≻ t s . Let F be a homogeneous polynomial of S of degree d such that F does notvanish on all X . By the division algorithm [1, Theorem 1.5.9, p. 30], we can write(3.2) F = h ( t q − − t q − s ) + · · · + h s − ( t q − s − − t q − s ) + F ′ , where F ′ is a homogeneous polynomial with deg t i ( F ′ ) ≤ q − i = 1 , . . . , s − F ′ ) = d .Let d ′ be the degree of the polynomial F ′ ( t , . . . , t s − , Z F ( t ,...,t s − , = { ( x , . . . , x s − , ∈ ( K ∗ ) s − × { } | F ( x , . . . , x s − ,
1) = 0 } ,A F = { [ x ] ∈ X | F ( x ) = 0 } . Notice that there is a bijection Z F ( t ,...,t s − , ψ −→ A F , ( x , . . . , x s − , ψ [( x , . . . , x s − , . Indeed ψ is clearly well defined and injective. To see that ψ is onto take a point [ x ] in A F with x = ( x , . . . , x s ). As F is homogeneous of degree d , form the equality F ( x /x s , . . . , x s − /x s ,
1) = F ( x , . . . , x s ) /x ds = 0 , we get that p = ( x /x s , . . . , x s − /x s ,
1) is a point in Z F ( t ,...,t s − , and ψ ( p ) = [ x ]. Hence | A F | = | Z F ( t ,...,t s − , | . Using Eq. (3.2), we get Z F ( t ,...,t s − , = Z F ′ ( t ,...,t s − , . We set H = H ( t , . . . , t s − ) = F ′ ( t , . . . , t s − ,
1) and Z H = { x ∈ ( K ∗ ) s − | H ( x ) = 0 } . The polynomial H does not vanish on ( K ∗ ) s − . This follows from Eq. (3.2) and using that F is homogeneous and that F does not vanish on X . We may assume that d ′ ≥
1, otherwise Z F ′ ( t ,...,t s − , = ∅ and | A F | = 0. Then, we can write d ′ = k ′ ( q −
2) + ℓ ′ for some integers k ′ ≥ ≤ ℓ ′ ≤ q −
2. Since k ≤ s −
2, by Lemma 3.3, we obtain that k ′ ≤ k and(3.3) − ( q − s − − k ′ + ℓ ′ ( q − s − − k ′ ≤ − ( q − s − − k + ℓ ( q − s − − k . Then, k ′ ≤ s − H is a non-zero polynomial of degree d ′ ≥ s − t i ( H ) ≤ q − i = 1 , . . . , s −
1. Therefore, applying Theorem 3.4 to H and then usingEq. (3.3), we derive | A F | = | Z H | ≤ ( q − s − k ′ − (( q − k ′ +1 − ( q −
1) + ℓ ′ ) ≤ ( q − s − k − (( q − k +1 − ( q −
1) + ℓ ) . Since F was an arbitrary homogeneous polynomial of degree d such that F does not vanish on X we obtain M := max {| A F | : F ∈ S d ; F } ≤ ( q − s − k − (( q − k +1 − ( q −
1) + ℓ ) , where F F is not the zero function on X . We claim that M = ( q − s − k − (( q − k +1 − ( q −
1) + ℓ ) . Let M be the expression in the right hand side. It suffices to show that M is bounded frombelow by M or equivalently it suffices to exhibit a homogeneous polynomial F d with exactly M roots in X . Let β be a generator of the cyclic group ( K ∗ , · ). Consider the ELISEO SARMIENTO, MARIA VAZ PINTO, AND RAFAEL H. VILLARREAL polynomial F = f f · · · f k g ℓ , where f , . . . , f k , g ℓ are given by f = ( βt − t )( β t − t ) · · · ( β q − t − t ) ,f = ( βt − t )( β t − t ) · · · ( β q − t − t ) , ... ... ... f k = ( βt − t k +1 )( β t − t k +1 ) · · · ( β q − t − t k +1 ) ,g ℓ = ( βt − t k +2 )( β t − t k +2 ) · · · ( β ℓ t − t k +2 ) . Now, the roots of F in X are in one to one correspondence with the union of the sets: { } × { β i } q − i =1 × ( K ∗ ) s − , { } × { } × { β i } q − i =1 × ( K ∗ ) s − , ... { } × · · · × { } × { β i } q − i =1 × ( K ∗ ) s − ( k +1) , { } × · · · × { } × { β i } ℓi =1 × ( K ∗ ) s − ( k +2) . Therefore the number of zeros of F in X is given by | A F | = ( q − q − s − + ( q − q − s − + · · · + ( q − q − s − ( k +1) + ℓ ( q − s − ( k +2) = ( q − s − ( k +2) h ( q − q − k + · · · + ( q − q −
1) + ℓ i = ( q − s − ( k +2) h ( q − q − q − k − + · · · + 1) + ℓ i = ( q − s − ( k +2) (cid:20) ( q − q − (cid:18) ( q − k − q − (cid:19) + ℓ (cid:21) = ( q − s − ( k +2) h ( q − k +1 − ( q −
1) + ℓ i , as required. Thus M = M and the claim is proved. Therefore δ d = min {k ev d ( F ) k : ev d ( F ) = 0; F ∈ S d } = | X | − max {| A F | : F ∈ S d ; F } = ( q − s − − (cid:16) ( q − s − k − (( q − k +1 − ( q −
1) + ℓ ) (cid:17) = ( q − s − k − (( q − − ℓ ) , where k ev d ( F ) k is the number of non-zero entries of ev d ( F ). This completes the proof of thecase 1 ≤ d ≤ ( q − s − −
1. Next we consider the case d ≥ ( q − s − δ d = 1 for d ≥ reg( S/I ( X )). Hence, applying Proposition 2.1, we get δ d = 1 for d ≥ ( s − q − (cid:3) The next proposition is an immediate consequence of our result. Recall that a linear code iscalled maximum distance separable (MDS for short) if equality holds in the Singleton bound.
Proposition 3.6. [11, 17] If X is a projective torus in P , then C X ( d ) is an MDS code and itsminimum distance is given by δ d = (cid:26) q − − d if ≤ d ≤ q − , if d ≥ q − . HE MINIMUM DISTANCE OF PARAMETERIZED CODES ON PROJECTIVE TORI 9 If X is a projective torus in P , then the minimum distance of C X ( d ) is given by δ d = ( q − − d ( q − if ≤ d ≤ q − , q − d − if q − ≤ d ≤ q − , if d ≥ q − . Parameterized codes arising from complete bipartite graphs have been studied in [10]. In thiscase one can use Theorem 3.5 and the next result to compute the minimum distance.
Theorem 3.7. [10]
Let K k,ℓ be a complete bipartite graph, let X be the toric set parameterizedby the edges of K k,ℓ , and let X and X be the projective torus of dimension ℓ − and k − respectively. Then, the length, dimension and minimum distance of C X ( d ) are equal to ( q − k + ℓ − , H X ( d ) H X ( d ) , and δ δ respectively, where δ i is the minimum distance of C X i ( d ) . Complete intersection ideals of parameterized sets of clutters
We continue to use the notation and definitions used in the introduction and in the prelim-inaries. In this section we characterize the ideals I ( X ) that are complete intersection when X arises from a clutter. Then, we show an optimal upper bound for the regularity of S/I ( X ). Definition 4.1. A clutter C is a family E of subsets of a finite ground set Y = { y , . . . , y n } such that if f , f ∈ E , then f f . The ground set Y is called the vertex set of C and E iscalled the edge set of C , they are denoted by V C and E C respectively.Clutters are special hypergraphs. One example of a clutter is a graph with the vertices andedges defined in the usual way for graphs.Let C be a clutter with vertex set V C = { y , . . . , y n } and let f be an edge of C . The charac-teristic vector of f is the vector v = P y i ∈ f e i , where e i is the i th unit vector in R n . Throughoutthis section we assume that v , . . . , v s is the set of all characteristic vectors of the edges of C .Recall that the algebraic toric set parameterized by y v , . . . , y v s , denoted by X , is the set X := { [( x v · · · x v n n , . . . , x v s · · · x v sn n )] ∈ P s − | x i ∈ K ∗ for all i } , where v i = ( v i , . . . , v in ) ∈ N n for i = 1 , . . . , s . Definition 4.2. If a ∈ R s , its support is defined as supp( a ) = { i | a i = 0 } . Note that a = a + − a − ,where a + and a − are two non-negative vectors with disjoint support called the positive and negative part of a respectively. Lemma 4.3. [21, Lemma 3.4]
Let C be a clutter. If f = 0 is a homogeneous polynomial of I ( X ) of the form t bi − t c with b ∈ N , c ∈ N s and i / ∈ supp( c ) , then deg( f ) ≥ q − . Moreover if b = q − , then f = t q − i − t q − j for some j = i .Proof. We may assume that f = t b − t c · · · t c r r , where c j ≥ j and b = c + · · · + c r . Then(4.1) ( x v · · · x v n n ) b = ( x v · · · x v n n ) c · · · ( x v r · · · x v rn n ) c r for all ( x , . . . , x n ) ∈ ( K ∗ ) n . We proceed by contradiction. Assume that b < q −
1. We claim that if v k = 1 for some1 ≤ k ≤ n , then v jk = 1 for j = 2 , . . . , r , otherwise if v jk = 0 for some j ≥
2, then making x i = 1for i = k in Eq. (4.1) we get ( x v k k ) b = x bk = x mk , where m < b . Then x b − mk = 1 for x k ∈ K ∗ .In particular if β is a generator of the cyclic group ( K ∗ , · ), then β b − m = 1. Hence b − m is amultiple of q − b ≥ q −
1, a contradiction. This completes the proof of the claim. Therefore supp( v ) ⊂ supp( v j ) for j = 2 , . . . , r . Since C is a clutter we get that v = v j for j = 2 , . . . , r , a contradiction because v , . . . , v r are distinct. Thus b ≥ q −
1. The second partof the lemma follows using similar arguments (see [21]). (cid:3)
A polynomial of the form f = t a − t b , with a, b ∈ N s , is called a binomial of S . The monomials t a and t b are called the terms of f . An ideal generated by binomials is called a binomial ideal . Theorem 4.4.
Let C be a clutter. If I ( X ) is a complete intersection, then I ( X ) = ( t q − − t q − s , . . . , t q − s − − t q − s ) . Proof.
According to [17, Theorem 2.1] the vanishing ideal I ( X ) is a binomial ideal. Notice that I ( X ) has height s −
1. Indeed, let [ P ] be an arbitrary point in X , with P = ( α , . . . , α s ), andlet I [ P ] be the ideal generated by the homogeneous polynomials of S that vanish at [ P ]. Then I [ P ] = ( α t − α t , α t − α t , . . . , α t s − α s t ) and I ( X ) = \ [ P ] ∈ X I [ P ] and the later is the primary decomposition of I ( X ), because I [ P ] is a prime ideal of S for any[ P ] ∈ X . As I [ P ] has height s − P ] ∈ X , we get that the height of I ( X ) is s −
1. As I ( X ) is a complete intersection of height s −
1, there is a minimal set B = { h , . . . , h s − } of homogeneous binomials that generate the ideal I ( X ). The set B is minimal in the sense that( B \ { h i } ) ( I ( X ) for all i . We may assume that h , . . . , h m are the binomials of B that containa term of the form t c i i . By Lemma 4.3 we have that deg( h i ) ≥ q − i = 1 , . . . , m . Thus wemay assume that h , . . . , h k are the binomials of B of degree q − t q − i and that h k +1 , . . . , h m have degree greater than q −
1. By Lemma 4.3 the binomials h , . . . , h k have the form t q − i − t q − j . Notice that ( I ( X ) : t i ) = I ( X ) for all i , this equality followsreadily using that t i does not vanish at any point of X . Hence, by the minimality of B , thebinomials h m +1 , . . . , h s − have both of their terms not in the set { t a , . . . , t a s s | a i ≥ i } .Since t q − i − t q − s is in I ( X ) for i = 1 , . . . , s −
1, we can write t q − i − t q − s = k X ℓ =1 λ ℓ h ℓ + m X ℓ = k +1 µ ℓ h ℓ + s − X ℓ = m +1 θ ℓ h ℓ ( λ ℓ , µ ℓ , θ ℓ ∈ S ) . As h , . . . , h s − are homogeneous binomials we can rewrite this equality as: t q − i − t q − s = k X ℓ =1 λ ′ ℓ h ℓ + s − X ℓ = m +1 θ ′ ℓ h ℓ , where λ ′ ℓ ∈ K for ℓ = 1 , . . . , k and for each m + 1 ≤ ℓ ≤ s − θ ′ ℓ = 0 and deg( h ℓ ) > q − h ℓ ) ≤ q − h ℓ ) + deg( θ ′ ℓ ) = q −
1. Then t q − i − t q − s − k X ℓ =1 λ ′ ℓ h ℓ = s − X ℓ = m +1 θ ′ ℓ h ℓ . The left hand side of this equality has to be zero, otherwise a non-zero monomial that occur inthe left hand side will have to occur in the right hand side which is impossible because monomialsoccurring on the left have the form λt q − j , λ ∈ K , and monomials occurring on the right arenever of this form. Hence we get the inclusion( t q − − t q − s , . . . , t q − s − − t q − s ) ⊂ ( h , . . . , h k ) . HE MINIMUM DISTANCE OF PARAMETERIZED CODES ON PROJECTIVE TORI 11
Since the height of ( h , . . . , h k ) is at most k , we get s − ≤ k . Consequently k = s −
1. Thusthe inclusion above is an equality as required. (cid:3)
Corollary 4.5.
Let C be a clutter with s edges and let T = { [( x , . . . , x s )] ∈ P s − | x i ∈ K ∗ } bea projective torus. The following are equivalent: (c ) I ( X ) is a complete intersection. (c ) I ( X ) = ( t q − − t q − s , . . . , t q − s − − t q − s ) . (c ) X = T ⊂ P s − .Proof. (c ) ⇒ (c ): It follows at once from Theorem 4.4. (c ) ⇒ (c ): By Proposition 2.1 one has I ( X ) = I ( T ) = ( { t q − i − t q − s } s − i =1 ). As X and T are both projective varieties, we get that X = T (see [17, Lemma 4.2] for details). (c ) ⇒ (c ): It follows at once from Proposition 2.1. (cid:3) The next result shows that the regularity of complete intersections associated to cluttersprovide an optimal bound for the regularity of
S/I ( X ). Proposition 4.6. reg(
S/I ( X )) ≤ ( q − s − , with equality if I ( X ) is a complete intersectionassociated to a clutter with s edges.Proof. For i ≥
0, we set h i = dim K ( S/ ( I ( X ) , t s )) i . Let r be the index of regularity of S/I ( X ).Then, h i > i = 0 , . . . , r and h i = 0 for i > r (see Section 2). Since t s does not vanish atany point of X , one has ( I ( X ) : t s ) = I ( X ). Therefore, there is an exact sequence of graded S -modules 0 −→ ( S/I ( X ))[ − t s −→ S/I ( X ) −→ S/ ( I ( X ) , t s ) −→ , where ( S/I ( X ))[ −
1] is the S -module with the shifted graduation such that( S/I ( X ))[ − i = ( S/I ( X )) i − for all i . Therefore from the exact sequence above we get h i = H X ( i ) − H X ( i − ≥ i ≥
1. On the other hand there is a surjection of graded S -modules D = S/ ( { t q − i − t q − s } s − i =1 ∪ { t s } ) = K [ t , . . . , t s − ] / ( { t q − i } s − i =1 ) −→ S/ ( I ( X ) , t s ) −→ . The Hilbert series of D is equal to the polynomial (1+ t + · · · + t q − ) s − because D is a completeintersection [29, p. 104]. Hence D i = 0 for i ≥ ( q − s −
1) + 1. From the surjection above weget that dim K D i ≥ h i ≥ i . If i ≥ ( q − s −
1) + 1, we obtain 0 = dim K D i ≥ h i ≥ H X ( i ) = H X ( i −
1) for i − ≥ ( q − s − . Hence reg(
S/I ( X )) ≤ ( q − s − I ( X ) is a completeintersection, then by Corollary 4.5 the ideal I ( X ) is equal to ( t q − − t q − s , . . . , t q − s − − t q − s ).Consequently reg( S/I ( X )) = ( q − s − (cid:3) Let X be an algebraic toric set parameterized by arbitrary monomials y v , . . . , y v s . A goodparameterized code should have large | X | and with dim K C X ( d ) / | X | and δ d / | X | as large as pos-sible. The following easy result gives an indication of where to look for non-trivial parameterizedcodes. Only the codes C X ( d ) with 1 ≤ d < reg( S/I ( X )) are interesting. Proposition 4.7. δ d = 1 for d ≥ reg( S/I ( X )) . Proof.
Since H X ( d ) is equal to the dimension of C X ( d ) and H X ( d ) = | X | for d ≥ reg( S/I ( X )),by a direct application of the Singleton bound we get that δ d = 1 for d ≥ reg( S/I ( X )). (cid:3) A well known general fact about parameterized linear codes is that the dimension of C X ( d )is strictly increasing, as a function of d , until it reaches a constant value. This behaviour waspointed out in [5] (resp. [8]) for finite (resp. infinite) fields. The minimum distance of C X ( d )has the opposite behaviour as the following result shows. Proposition 4.8. [17, 26] If δ d > resp. δ d = 1) , then δ d > δ d +1 ( resp. δ d +1 = 1) . ACKNOWLEDGMENTSThe authors would like to thank two anonymous referees for providing us with useful commentsand suggestions, and for pointing out that Proposition 4.8 was first shown by S. Tohˇaneanu in [26,Proposition 2.1]. The authors would also like thank Hiram L´opez, who provided an alternativeproof of Theorem 3.5, and Carlos Renter´ıa for many stimulating discussions.
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