Standard Bases for Linear Codes over Prime Fields

Transcription

1 Standard Bases for Linear Codes over Prime Fields arxiv: v1 cs.it] 18 Aug 2017 Jean Jacques Ferdinand RANDRIAMIARAMPANAHY 1 randriamiferdinand@gmail.com Harinaivo ANDRIATAHINY 2 hariandriatahiny@gmail.com Toussaint Joseph RABEHERIMANANA 3 rabeherimanana.toussaint@yahoo.fr 1,2,3 Mention : Mathematics and Computer Science, Domain : Sciences and Technologies, University of Antananarivo, Madagascar August 21, 2017 Abstract It is known that a linear code can be represented by a binomial ideal. In this paper, we give standard bases for the ideals in a localization of the multivariate polynomial ring in the case of the linear codes over prime fields. Keywords : Linear code, semigroup order, Groebner basis, local ring, standard basis MSC 2010 : 13P10, 94B05, 12E20 1 Introduction Coding theory is the mathematical basis for data transmission through noisy communication channels. It contains two main parts. The first part is to encode the message to reduce its sensitivity to noise during transmission. The second part is to decode the received message by detecting and correcting the errors. Bruno Buchberger introduced the theory of Groebner bases for polynomial ideals in The Groebner bases theory can be used to solve some problems concerning the ideals by developing computations in multivariate polynomial rings. In 1964, Hironaka 12] introduced the analogues of Groebner bases called standard bases for ideals in the localization of the polynomial ring at the origin. In 6], standard bases for ideals generated by polynomials in local rings can be determined by using the same method as Groebner bases. Connection between linear codes and ideals in polynomial rings was presented in 2]. And it was proved that a Groebner basis of the ideal associated to a binary linear code can be used for determining the minimum distance. A generalization to linear codes over prime fields can be found in 15, 16]. In 15], ias been proved that a linear code can be described by a binomial ideal, and a Groebner basis with respect to lexicographic order for the binomial ideal is determined. 1

2 The aim of this paper is to present the standard basis of the ideal of a linear code over a prime field in the local ring of rational fonctions that are regular at a point of the affine variety associated to the ideal. The idea is to generalize the method developed by N. Dück and K. H. Zimmermann in 9]. This paper is organized as follows. The second section presents a background for Groebner bases. The third section contains the division algorithm in a local ring. In section 4, the notion of linear codes and their connections with binomial ideals are presented. The main result is contained in section 5. 2 Preliminaries Throughout this paper, n denotes a positive integer, K a commutative field and KX] := KX 1,...,X n ] the polynomial ring in n variables over K. We denote by 0 the zero element of N n where N is the set of non negative integers. A monomial in KX] is an algebraic expression of the form X α 1 1 Xn αn which is denoted by X α where α = α 1,...,α n N n. The monomial X α = X α 1 1 X αn n can be identified with the n-tuple of exponents α = α 1,...,α n N n and vice versa, thus there exists a one-to-one correspondence between the monomials in KX] and the elements of N n. The degree of the monomial X α with α = α 1,...,α n is α := α 1 +α 2 + +α n. Any order > we establish on N n will give us an order on the set of monomials in KX] : if α > β according to this order, we have X α > X β. An order > in KX] is compatible with multiplication if for all X α, X β and X γ in KX] with X α > X β then X α X γ > X β X γ. Now let > be an order on the set of monomials X u where u N n. We say that > is a semigroup order in KX] if > is a total order and it is compatible with the multiplication of monomials. A monomial order on KX] is a semigroup order such that 1 < X i for i = 1,...,n. Usual monomial orders on KX] are the lexicographic order, the degree lexicographic order and the degree reverse lexicographic order. Let f be a non-zero polynomial of KX] such that f = k c i X α i where c i K and α i N n. Let us fix a monomial order > on KX]. A term in KX] is a scalar times a monomial. The leading term of f, denoted by lt > f, is the largest involved term with respect to >. If lt > f = c m X αm where 1 m k, then c m is called the leading coefficient of f lc > f and X αm is the leading monomial of f lm > f. We denote by degf the maximal degree of all monomials occuring in f. The reduction of a polynomial f by a polynomial g, denoted by Redf,g is defined by i=1 Redf,g := f q.g wherelt > f = q.lt > g, for some termq = cx α. LetF = f 1,...,f s be a s-tuple of polynomials in KX]. Each polynomial f KX] can be written in the form : f = a 1 f 1 +a 2 f 2 + +a s f s +r where a 1,...,a s, r KX] and either r = 0 or r is a K-linear combination of monomials, none of which is divisible by any of lt > f 1,...,lt > f s. Moreover, if a i f i 0, then lt > f lt > a i f i, 1 i s. The polynomial r is called the remainder of f on division by F. The remainder r is produced by the following algorithm called division algorithm in KX] see 5]. Input : f 1,..., f s, f Output : r r := 0 p := f WHILE p 0 DO 2

3 i := 1 divisionoccurred:= false WHILE i s AND divisionoccurred = false DO IF lt > f i divides lt > p THEN p := Redp,f i divisionoccurred:= true ELSE i := i+1 IF divisionoccurred = false THEN r := r +lt > p p := p lt > p The division algorithm terminates after a finite number of steps. Let I KX] be a non-zero ideal and > a monomial order. The ideal generated by the set of lt > f where f I is called the leading ideal of I, denoted by lt > I, i.e lt > I := lt > f/f I. For a finite subset G = {g 1,...,g s } of the ideal I, we denote by lt > G the ideal generated by the lt > g i, for i = 1,...,s, i.e lt > G := lt > g 1,...,lt > g s. A finite subset G = {g 1,...,g s } of an ideal I is called a Groebner basis of I if lt > G = lt > I. A Groebner basis G for the polynomial ideal I is called a minimal Groebner basis of I if for all g G, lc > g = 1 and lt > g lt > G {g}. The reduced Groebner basis for the ideal I is a Groebner basis G satisfying : 1 lc > g = 1 for all g G, 2 no monomial of g lies in lt > G {g}, for all g G. A Groebner basis can be determined by using Buchberger s algorithm. Let f, g KX] be non-zero polynomials. Fix a monomial order > and let X γ be the least common multiple of the leading monomial of f and the leading monomial of g. The S-polynomial of f and g, denoted by spolyf, g is spolyf,g := Xγ Xγ.f lt > f lt > g.g If I is a polynomial ideal, then a basis G = {g 1,...,g s } for I is a Groebner basis for I if and only if for all pairs i, the remainder on division of spolyg i,g by G is zero. 3 Localization Let K be a commutative field and p = p 1,...,p n an n-tuple of K n. KX := KX 1,...,X n denotes the field { of rationnal fonctions containing } KX]. We define a local ring in KX by f the set O p := /f,g KX],gp 0. We say also that we localize at the maximal ideal g X 1 p 1,...,X n p n KX]. Let m p be the ideal generated by X 1 p 1,...,X n p n in O p. Then each element in O p \m p is a unit in O p. A local order in KX] = KX 1,...,X n ] is a semigroup order such that 1 > X i for all 1 i n. For instance, for two n-tuples of N n α = α 1,...,α n and β = β 1,...,β n, we define the negative degree lexicographic order by α > β if α < β or α = β and there exists an integer i {1,...,n} such that α 1 = β 1,...,α i 1 = β i 1,α i > β i. 3

4 Let > be a local order on the set of monomials of KX] and let S > := {1+g KX]/ g = 0 or lt > g < 1}. S > is a multiplicative part of KX] and we remark that S > = KX]\ X 1,...,X n. Define the localization of KX] in view of the order > by the ring 6, 13] { } f Loc > KX] := S> 1 KX] = 1+g /f KX], 1+g S >. 1 We have Loc > KX] = O p=0. Under the local order, there is a difficulty for the successive reductions, because we may have an infinite strictly decreasing sequence of terms. For example, consider the polynomials of one variable X, f = X and g = X X 2, and we divide f by g by using the division algorithm, so that we successively reduce by X X 2. This gives the reductions: f 1 := Redf,g = X 2 f 2 := Redf 1,g = X 3. f n := Redf n 1,g = X n+1, and so on. Mora introduced a method to solve this problem. The following result can be found in 6]. Theorem 3.1 Mora normal form algorithm. Given non-zero polynomials f,f 1,...,f s KX] and let > be a local order. There is an algorithm which gives the polynomials u,a 1,...,a s,h KX] such that uf = a 1 f 1 + +a s f s +h 2 where lt > u = 1 u = 1 + g is unit in Loc > KX], lt > f lt > a i lt > f i for all i with a i 0, and h = 0 or lt > h is not divisible by any of lt > f i. We denote NFf G := h with G = {f 1,...,f s }, and we say tha is the weak normal form of f on division by G. For f KX], we define ecartf := degf deglt > f. The remainder h in 2 is produced by the following algorithm called Mora s division algorithm h := f; L := {f 1,...,f s }; M := {g L : lt > g lt > h} WHILE h 0 AND M DO SELECT g M with ecartg minimal IF ecartg > ecarth THEN L := L {h} h := Redh,g IF h 0 THEN M := {g L : lt > g lt > h} 4 Linear codes and binomial ideals Let F p be the finite field with p elements where p is a prime number. A linear code C of length n and dimension k over F p is the image of a linear inective mapping ψ : F k p Fn p 4

5 where k n. The elements x = x 1,...,x n C are called the codewords. The weight of a word x = x 1,...,x n F n p is defined by w tx := card{i/x i 0,1 i n}. The minimum distance of the linear code C is d := min{dx,y/x,y C,x y} or d := min{w t x/ x C,x 0} where dx,y := card{i/x i y i }. We define the support of an element x C by suppx := {i/x i 0}. A linear code C of length n and dimension k is called an n,k] code. Moreover, if the minimum distance is d, we say that C is an n,k,d] code. Let C be an n,k] code, e i = ζ i1,...,ζ ik where i = 1,...,k the canonical basis of F k p and ψe i = g i1,...,g in. The generating matrix of C is the matrix of dimension k n defined by G = g i where g i F p. The linear code C is represented as follows C = {xg/ x F k p }. We will say that G is in standard form if G = I k M where I k is the k k identity matrix. Let C be an n,k] code over F p. Define the ideal associated with C as 2, 8, 10, 16] I C := X c X c c c C + X p i 1 1 i n 3 where each word c F n p is considered as an integral vector in the monomial Xc. Let C be an n,k] code over F p and G = g i = I k M 4 a generating matrix in standard form. Let m i be the vector of length n over F p defined by m i = 0,...,0,p g i,k+1,...,p g i,n 5 for 1 i k. We have X m i = X p g i,k+1 k+1 X p g i,k+2 k+2...x p g i,n n = if suppm i =, then X m i = 1. suppm i Theorem 4.1. Let us take the lexicographic order on KX] with X 1 > X 2 > > X n. The code ideal I C has the reduced Groebner basis X p g i,. In particular, G = {X i X m i /1 i k} {X p i 1/k +1 i n}. 6 Proof. A proof can be found in 15]. 5 Standard bases In this section, we will describe the analogues of Groebner bases called standard bases for the ideals in local rings by Mora s division algorithm. Given any local order > on monomials in KX], there is a natural extension of > to Loc > KX], which we will also denote by >. For any h = f 1+g Loc >KX] as in 1, we define lm > h := lm > f; lc > h := lc > f and lt > h := lt > f. We fix a local order > on Loc > KX] and let I be an ideal in Loc > KX]. A standard basis of I is a subset {f 1,...,f r } of I such that lt > I = lt > f 1,...,lt > f r where lt > I is the ideal generated by the set of lt > f with f I. Proposition 5.1 Product criterion. 11] Let f,g KX 1,...,X n ] be polynomials such that lcmlm > f,lm > g = lm > f.lm > g, then NF spolyf,g {f,g} = 0 7 where NF is defined as in theorem3.1. We will consider the ideals of the local ring Loc > KX] which are generated by polynomials of KX]. A more general result of the following theorem can be found in 11]. 5

6 Theorem 5.2 Buchberger criterion. Let I Loc > KX] be an ideal, G = {g 1,...,g s } a set of polynomials of I and > a local order. Let NF be the weak normal form as in theorem3.1. Then the following are equivalent: i G is a standard basis of I. ii G generates I and NF spolyg i,g G = 0 for i, = 1,...,s. iii G generates I and NF spolyg i,g G i = 0 for a suitable subset Gi G and i, = 1,...,s. Let C be an n,k] code, the point 1,...,1 is a zero of the code ideal I C in the affine space over F p. Rather than localizing at the maximal ideal X 1 1,...,X n 1, we change coordinates to translate the point to the origin. Denote I C the corresponding ideal, and I := I C Loc >F p X] the ideal of Loc > F p X] generated by I C. Lemma 5.3. Let p a prime number, we have I C X = i +1+p 1 X +1 p g i, /1 i k + X i +1 p +p 1/k+1 i n suppm i where g i, is defined in 4 and m i in 5. Proof. The ideal I C defined in 3 has the reduced Groebner basis 6 by the theorem4.1 with respect to the lexicographic order on KX]. This is an ideal basis of I C in KX]. The translation is made via the ring map X i X i +1. Since KX] Loc > KX], then the claim for the translated ideal follows. Now we present our main result Theorem 5.4. Let C be an n,k] code over F p with p a prime number. Under the negative degree lexicographic order > on F p X], the ideal I = I C Loc >F p X] in Loc > F p X] has the standard basis S = { X i 0 t l p g i,l 1 l σ i t 1,...,t σi 0,...,0 } } X /1 i k {X pi /k +1 i n 8 where σ i := cardsuppm i. Proof. We will show that S generates I C, then we prove that S is a standard basis. For k +1 i n Let X p i S and we will prove that X p i I C. For {1,2,...,p 1}, the number p is a 6

7 multiple of p because p is prime. Since we work over a field of characteristic p, we have p 1 p X p i = X i +Xp i For 1 i k Let X i 0 t l p g i,l 1 l σ i t 1,...,t σi 0,...,0 =1 p 1 p = 1+ X i +Xp i 1 =1 p 1 p = X 0i 0 + p p X i + X p i p 1 =1 p p = X i 1 =0 p p = X i +p 1 =0 = X i +1 p +p 1 I C. with 1 < 2 < < σi and denote A := X i A = X i +1 = X i +1 = X i +1 = X i +1 X S. Suppose that suppm i = { 1, 2,..., σi } 0 t l p g i,l 1 l σ i t 1,...,t σi 0,...,0 t 1 =0 t 2 =0 t σi =0 p gi,1 0 t l p g i,l 1 l σ i t 1,...,t σi 0,...,0 X X +1 0 t l p g i,l 1 l σ i p g i,1 p g i,2 p g i,σi p gi,1 p gi,2 p g i,1 t 1 =0 t 1 t 1 ]p gi,2 X t 1 1 t 2 =0 = X i +1 X1 +1 p g i,1 ] X2 +1 p g i,2 = X i +1 = X i +1+p suppm i suppm i = X i +1+p 1 X +1 p g i, t 2 p gi,2 X +1 p g i, ] suppm i t 2 suppm i ] X +1 p g i, I C By lemma 5.3, S is a generating set for I C. Let us now show that S is a standard basis over F p 7 ] X. We have p gi,σi... X t 1 t 1 X t X tσ i σi σi ] p g i,σi X t t σi =0 ] ]... Xσi +1 p g i,σi X +1 p g i, p gi,σi t σi ] X tσ i σi

8 * Let the pair i, such that k +1 i < n. We havespolyx p i,xp = Xp i Xp X p i X p i Xp i Xp X p X p = Xp i Xp Xp i Xp = 0, andnf0 S = 0. * Let the pair i, such that 1 i k and k +1 n. Denote f i := X i X. We have spolyf i, X p = X ix p X i 0 t l p g i,l 1 l σ i t 1,...,t σi 0,...,0 X i = X i X p Xp = X p 0 t l p g i,l 1 l σ i t 1,...,t σi 0,...,0 0 t l p g i,l 1 l σ i t 1,...,t σi 0,...,0 0 t l p g i,l 1 l σ i t 1,...,t σi 0,...,0 ] X ] X X ix p X p X p X ] X i X p S. Therefore the remain- In the last expression, all these monomials are multiple of X p der of the division of spolyf i,x p by {Xp } is zero, i.e NF spolyf i,x p {Xp } = 0. * Finally, let 1 i < k. Let g i := X i and g := X 0 t l p g i,rl 1 l σ i t 1,...,t σi 0,...,0 0 t l p g,s l 1 l σ t 1,...,t σ 0,...,0 p gi,rh σ u=1 p g,su t u X rh X t u su. We have lt > g i = X i and lt > g = X, then lcmlm > g i,lm > g = lm > g i.lm > g. According to the Product Criterion in Proposition 5.1, we obtain NF spolyg i,g {g i,g } = 0. And by the Buchberger s criterion in theorem 5.2, the assertion follows. Example 5.5. Consider the generator matrix G = g i defined by G = Under the negative degree lexicographic order > on F 3 X], the ideal I = I C Loc >F 3 X] where I C is defined in lemma 5.3 with p = 3, n = 6 and k = 3 has the standard basis 8

9 { } S = g 1, g 2, g 3, X4, 3 X5, 3 X6 3 where g 1 = X 1 +X 4 +X 6 +2X X 4 X 6 +2X 2 6 +X 2 4X 6 +X 4 X X 2 4X 2 6, g 2 = X 2 +2X 4 +X 5 +X 4 X 5 +2X X 4X 2 5, g 3 = X 3 +2X 4 +2X 5 +X 6 +2X 4 X 5 +X 4 X 6 +X 5 X 6 +2X 2 6 +X 4X 5 X 6 +2X 4 X X 5X X 4X 5 X 2 6. An immediate consequence is the result in 9] for p = 2 Theorem 5.6. In view of the negative degree lexicographic order > on F 2 X], the ideal I = I C Loc >F 2 X] in Loc > F 2 X] where I C = X i +1+ X +1/1 i k J suppm i J suppm i + X i /k+1 i n has the standard basis : { S = X i } { } X J /1 i k Xi/k 2 +1 i n. References 1] W. Adams and P. Loustaunau, An Introduction to Groebner Bases, American Mathematical Society, Vol.3, ] M. Borges-quintana, M. Borges-trenard, P. Fitzpatrick and E. Martinez-moro, Groebner bases and combinatorics for binary codes, Applicable Algebra in Engineering Communication and Computing - AAECC, Vol.19, 2008, pp ] B. Buchberger, An Algorithm for Finding the Basis Elements of the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal, PhD thesis, University of Innsbruck, ] A.Cooper, Towards a new method of decoding Algebraic codes using Groebner bases, Transactions 10th Army Conf. Appl. Math. Comp., Vol.93, 1992, pp ] D. Cox, J. Little and D. O Shea, Ideals, Varieties, and Algorithms, Springer, ] D. Cox, J. Little and D. O Shea, Using Algebraic Geometry, Springer, ] N. Dück and K.-H. Zimmermann, Graver Bases and Universal Gröbner Bases for Linear Codes, May ] N. Dück and K.-H. Zimmermann, Gröbner bases for perfect binary linear codes. International Journal of Pure and Applied Mathematics, Vol 91, 2014, ] N. Dück and K.-H. Zimmermann, Standard Bases for binary Linear Codes, International Journal of Pure and Applied Mathematics Volume 80, ] N. Dück and K.-H. Zimmermann, Universal Groebner bases for Binary Linear Code, International Journal of Pure and Applied Mathematics, Appr ] G.-M. Greuel and G. Pfister. A Singular Introduction to Commutative Algebra, Springer- Verlag, Berlin,

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