Generalization of the Lee-O Sullivan List Decoding for One-Point AG Codes

Transcription

1 arxiv: v2 [cs.it] 5 Apr 2012 Generalization of the Lee-O Sullivan List Decoding for One-Point AG Codes Olav Geil Ryutaroh Matsumoto Diego Ruano April 5, 2012 Abstract We generalize the list decoding algorithm for Hermitian codes proposed by Lee and O Sullivan [8] based on Gröbner bases to general one-point AG codes, under an assumption weaker than one used by Beelen and Brander [2]. Our generalization enables us to apply the fast algorithm to compute a Gröbner basis of a module proposed by Lee and O Sullivan [8], which was not possible in another generalization by Lax [7]. Keywords: algebraic geometry code, Gröbner basis, list decoding 1 Introduction We consider the list decoding of one-point algebraic geometry (AG) codes. Guruswami and Sudan [5] proposed the well-known list decoding algorithm for onepoint AG codes, which consists of the interpolation step and the factorization step. The interpolation step has large computational complexity and many researchers This research was partly supported by the MEXT Grant-in-Aid for Scientific Research (A) No , the Villum Foundation through their VELUX Visiting Professor Programme , the Danish National Research Foundation and the National Science Foundation of China (Grant No ) for the Danish-Chinese Center for Applications of Algebraic Geometry in Coding Theory and Cryptography and by Spanish grant MTM The proposed algorithm in this paper is submitted without any proof of its correctness for possible presentation at 2012 IEEE International Symposium on Information Theory, Boston, MA, USA, July Department of Mathematical Sciences, Aalborg University, Denmark Department of Communications and Integrated Systems, Tokyo Instiutte of Technology, Japan 1

2 have proposed faster interpolation steps, see [2, Figure 1]. Lee and O Sullivan [8] proposed a faster interpolation step based on the Gröbner basis theory for onepoint Hermitian codes. Beelen and Brander [2] proposed the fastest interpolation procedure for the so-called C ab curves [12] with an additional assumption [2, Assumptions 1 and 2]. Little [9] generalized the method in Lee and O Sullivan [8] by using the same assumption as Beelen and Brander [2, Assumptions 1 and 2]. Lax [7] generalized part of [8] to general algebraic curves, but he did not generalize the faster interpolation algorithm in [8]. The aim of this paper is to generalize the faster interpolation algorithm [8] to an even wider class of algebraic curves than [9]. We shall demonstrate that our proposal provides much faster alternative to the previously known interpolation step for the code on the Klein quartic in Example 12. As a byproduct of our argument, in Corollary 7 we also clarifies the relation between two different definitions of modules used by Sakata [16] and by Lax [7], Lee and O Sullivan [8] for list decoding. This paper is organized as follows: Section 2 introduces notations and relevant facts. Section 3 generalizes [8]. Section 4 concludes the paper. 2 Notation and Preliminary Our study heavily relies on the standard form of algebraic curves introduced independently by Geil and Pellikaan [4] and Miura [13], which is an enhancement of earlier results [12, 15]. Let F/F q be an algebraic function field of one variable over a finite field F q with q elements. Let g be the genus of F. Fix n+1 distinct places Q, P 1,..., P n of degree one in F and a nonnegative integer u. We consider the following one-point algebraic geometry (AG) code C u ={( f (P 1 ),..., f (P n )) f L(uQ)}. Suppose that the Weierstrass semigroup H(Q) at Q is generated by a 1,..., a t, and choose t elements x 1,..., x t in F whose pole divisors are (x i ) = a i Q for i=1,..., t. We do not assume that a 1 is the smallest among a 1,..., a t. Without loss of generality we may assume the availability of such x 1,..., x t, because otherwise we cannot find a basis of C u for every u. Then we have thatl( Q)= i=1 L(iQ) is equal to F q[x 1,..., x t ] [15]. We expressl( Q) as a residue class ring F q [X 1,..., X t ]/I of the polynomial ring F q [X 1,..., X t ], where X 1,..., X t are transcendental over F q, and I is the kernel of the canonical homomorphism sending X i to x i. Geil and Pellikaan [4] and Miura [13] identified the following convenient representation of L( Q) by using the Gröbner basis theory [1]. The 2

3 following review is borrowed from [11]. Hereafter, we assume that the reader is familiar with the Gröbner basis theory in [1]. Let N 0 be the set of nonnegative integers. For (m 1,..., m t ), (n 1,..., n t ) N t 0, we define the weighted reverse lexicographic monomial order such that (m 1,..., m t ) (n 1,..., n t ) if a 1 m a t m t > a 1 n 1 + +a t n t, or a 1 m a t m t = a 1 n 1 + +a t n t, and m 1 = n 1, m 2 = n 2,..., m i 1 = n i 1, m i < n i, for some 1 i t. Note that a Gröbner basis of I with respect to can be computed by [15, Theorem 15] or [18, Proposition 2.17], starting from any defining equations of F/F q. Example 1 According to Høholdt and Pellikaan [6, Example 3.7], u 3 v+v 3 + u=0 is an affine defining equation for the Klein quartic over F 8. There exists a unique F 8 -rational place Q such that (v) = 3Q, (uv) = 5Q, and (u 2 v) = 7Q. The numbers 3, 5 and 7 constitute the minimal generating set of the Weierstrass semigroup at Q. Choosing v as x 1, uv as x 2 and u 2 v as x 3, by Vasconcelos [18, Proposition 2.17] we can see that the standard form of the Klein quartic is given by X X 3X 1, X 3 X 2 + X X 2, X X 2X X 3, which is the reduced Gröbner basis with respect to the monomial order. We can see that a 1 = 3, a 2 = 5, and a 3 = 7. For i=0,..., a 1 1, we define b i = min{m H(Q) m i (mod a 1 )}, and L i to be the minimum element (m 1,..., m t ) N t 0 with respect to such that a 1 m 1 + +a t m t = b i. Note that the set of b i s is the well-known Apéry set [14, Lemmas 2.4 and 2.6] of the numerical semigroup H(Q). Then we have l 1 = 0 if we write L i as (l 1,...,l t ). For each L i = (0,l i2,...,l it ), define y i = x l i2 2 xl it t L( Q). The footprint of I, denoted by (I), is{(m 1,..., m t ) N t 0 Xm 1 1 X m t t is not the leading monomial of any nonzero polynomial in I with respect to }, and define B={x m 1 1 xm t t (m 1,..., m t ) (I)}. Then B is a basis ofl( Q) as an F q -linear space [1], two distinct elements in B has different pole orders at Q, and B = {x m 1 xl 2, 2 xl t t m N 0, (0,l 2,...,l t ) {L 0,..., L a1 1}} = {x m 1 y i m N 0, i=0,..., a 1 1}. (1) Equation (1) shows thatl( Q) is a free F q [x 1 ]-module with a basis{y 0,..., y a1 1}. Note that the above structured shape of B reflects the well-known property of every weighted reverse lexicographic monomial order, see the paragraph preceding to [3, Proposition 15.12]. 3

4 Example 2 For the curve in Example 1, we have y 0 = 1, y 1 = x 3, y 2 = x 2. Let v Q be the unique valuation in F associated with the place Q. The semigroup H(Q) is equal to{ia 1 v Q (y j ) 0 i, 0 j<a 1 } [14, Lemma 2.6]. 3 Generalization of Lee-O Sullivan s List Decoding to General One-Point AG Codes 3.1 Background on Lee-O Sullivan s Algorithm In the famous list decoding algorithm for the one-point AG codes in [5], we have to compute the univariate interpolation polynomial whose coefficients belong to L( Q). Lee and O Sullivan [8] proposed a faster algorithm to compute the interpolation polynomial for the Hermitian one-point codes. Their algorithm was sped up and generalized to one-point AG codes over the so-called C ab curves [12] by Beelen and Brander [2] with an additional assumption. In this section we generalize Lee-O Sullivan s procedure to general one-point AG codes with an assumption weaker than [2, Assumption 2], which will be introduced in and used after Assumption 9. The argument before Assumption 9 is true without Assumption 9. Let m be the multiplicity parameter in [5]. Lee and O Sullivan [8] introduced the ideal I r,m for Hermitian curves containing the interpolation polynomial corresponding to the received word r and the multiplicity m. The ideal I r,m contains the interpolation polynomial as its nonzero element minimal with respect to the weighted reverse lexicographic monomial order u to be introduced in Section 3.3. We will give a generalization of I r,m for general algebraic curves. 3.2 Generalization of the Interpolation Ideal Let r=(r 1,..., r n ) F n q be the received word. For a divisor G of F, we define L( G+ Q)= i=1l( G+ iq). We see thatl( G+ Q) is an ideal ofl( Q) [10]. Let h r L( Q) such that h r (P i )=r i. Computation of such h r can be easily done as follows provided that we can construct generator matrices for C u for all u. We can compute Ĥ(Q)={u H(Q) C u C u 1 }. For each u Ĥ(Q), define 4

5 ψ u B such that v Q (ψ u )=u, and let i 1. i n ψ 1 (P 1 ) ψ 1 (P n ) =... ψ n (P 1 ) ψ n (P n ) 1 r. We find that h r = n j=1 i j ψ j satisfies the required condition for h r. Since max Ĥ(Q) n+2g 1, we can choose h r so that v Q (h r ) n+2g 1. Let Z be transcendental overl( Q), and D=P 1 + +P n.l( Q)[Z] denotes the univariate polynomial ring of Z overl( Q). For a divisor G we denote by L Z ( G+ Q) the ideal ofl( Q)[Z] generated byl( G+ Q) L( Q). Define the ideal I r,m ofl( Q)[Z] as I r,m = L Z ( md+ Q)+L Z ( (m 1)D+ Q) Z h r + +L Z ( D+ Q) Z h r m 1 + Z h r m, (2) where denotes the ideal generated by, the plus sign+denotes the sum of ideals, andl Z ( id+ Q) Z h r m i denotes the product of two idealsl Z ( id+ Q) and Z h r m i. We remark that the above I r,m is equivalent to Ī m,v by Lax [7]. Note that our definition does not involve coordinate variables x 1, x 2,... of the defining equations as used by Lax [7]. For Q(Z) L( Q)[Z], we say Q(Z) has multiplicity m at (P i, r i ) if Q(Z+ r i )= α j Z j (3) withα j L( Q) satisfies v Pi (α j ) m j for all j. Sakata [16, Section 3.2] introduced a special case of the following set for Hermitian curves. We give a more general definition as follows: I r,m ={Q(Z) L( Q)[Z] Q(Z) has multiplicity m for all (P i, r i )}. This definition of the multiplicity is the same as [5]. Therefore, we can find the interpolation polynomial used in [5] from I. We shall explain how to find efficiently the interpolation polynomial from I, after clarifying the relation between r,m r,m I r,m and I. r,m Lemma 3 We have I r,m I r,m. j 5

6 Proof. Observe that I r,m is an ideal ofl( Q)[Z]. Letα(Z h r) j L Z ( (m j)d+ Q) Z h r j such thatα L( (m j)d+ Q). Then we have α(z+ r i h r ) j =α(z (h r r i )) j = j α k (h r r i ) j k Z k, k=0 whereα k L( (m j)d+ Q). We can see thatα k (h r r i ) j k L( (m k)p i + Q) and thatl( (m j)d+ Q) Z h r j I, becausel r,m Z( (m j)d+ Q) Z h r j is generated by{α(z h r ) j α L( (m j)d+ Q)} as an ideal ofl( Q)[Z]. Since I is an ideal, it follows that I r,m r,m I. r,m The following Proposition 4 will be used in the proof of Proposition 6. Proposition 4 [5] dim Fq L( Q)[Z]/I r,m = n( m+1 2 Lemma 5 Let G be a divisor 0 whose support is disjoint from Q. If deg P=1 for all P supp(g) then we have dim Fq L( Q)/L( G+ Q)=deg G. Proof. Let n() be a mapping from supp(g) to the set of nonnegative integers. Let N be the set of those functions such that n(p)<v P (G) for all P supp(g). By the strong approximation theorem [17, Theorem I.6.4] we can choose a f n() L( Q) such that v P ( f n() )=n(p) for every P supp(g). Any element inl( Q)\L( G+ Q) can be written as the sum of an element g L( G+ Q) plus an F q -linear combination of f n() s by the assumption deg P=1 for all P supp(g), which completes the proof. The following proposition is equivalent to Lax [7, Proposition 6], but we include its proof because our definition of I r,m is apparently very different from that of Ī m,v by Lax [7]. Proposition 6 dim Fq L( Q)[Z]/I r,m = n ( m+1 2 Proof. Recall that I is an ideal of F q [X 1,..., X t ] such thatl( Q) = F q [X 1,..., X t ]/I as introduced in Section 2. Let G i be a Gröbner basis of the preimage of L( id+ Q) in F q [X 1,..., X t ], and H r be the coset representative of h r written as a sum of monomials whose exponents belong to (I). In this proof, the footprint ( ) is always considered for F q [X 1,..., X t ] excluding the variable Z. Then G= m i=0 {F(Z H r) m i F G i } 6 ). ).

7 is a Gröbner basis of the preimage of I r,m in F q [Z, X 1,..., X t ] with the elimination monomial order with Z greater than X i s and refining the monomial order defined in Section 2. Please refer to [3, Section 15.2] for refining monomial orders. A remainder of division by G can always be written as F m 1 Z m 1 + F m 2 Z m F 0 with F i F q [X 1,..., X t ]. Then F i must be written as a sum of monomials whose exponents belong to the footprint (G i ) of G i. This shows that On the other hand, by Lemma 5, This implies m 1 dim Fq L( Q)[Z]/I r,m (G i ). i=0 (G i )=dim Fq L( Q)/L( id+ Q)=ni. ( ) m+1 dim Fq L( Q)[Z]/I r,m n. 2 By Proposition 4 and Lemma 3, we see ( ) m+1 dim Fq L( Q)[Z]/I r,m = n. 2 The following corollary clarifies the relation between the module I r,m used by Sakata [16] and I r,m used by Lax [7], Lee and O Sullivan [8], which was not explicit in previous literature. Corollary 7 I r,m = I r,m. Since I is the ideal used in [5], we can find the required interpolation polynomial r,m directly from an F q [x 1 ]-submodule of I r,m = I as explained in Section 3.3. r,m For i=0,..., m and j=0,..., a 1 1, letη i, j to be an element inl( id+ Q) such that v Q (η i, j ) is the minimum among{η L( id+ Q) v Q (η) j (mod a 1 )}. Such elementsη i, j can be computed by [10] before receiving r. It was also shown [10] that{η i, j j=0,..., a 1 1} generatesl( id+ Q) as an F q [x 1 ]- module. Note also that we can chooseη 0,i = y i defined in Section 2. By Eq. (1), allη i, j and h r can be expressed as polynomials in x 1 and y 0,..., y a1 1. Thus we have 7

8 Theorem 8 (Generalization of Beelen and Brander [2, Proposition 6] and Little [9]) Letl m. One has that generates as an F q [x 1 ]-module. {(Z h r ) m i η i, j i=0,..., m, j=0,..., a 1 1} {Z l m (Z h r ) m η 0, j l=1,..., j=0,..., a 1 1} I r,m,l = I r,m {Q(Z) L( Q)[Z] deg Z Q(Z) l} Proof. Let e I r,m and E be its preimage in F q [Z, X 1,..., X t ]. By dividing E by the Gröbner basis G introduced in the proof of Proposition 6, we can see that e is expressed as m e= α l Z l (Z h r ) m + α i (Z h r ) m i l=1 withα i L( max{i, 0}D+ Q), from which the assertion follows. 3.3 Computation of the Interpolated Polynomial from the Interpolation Ideal I r,m For (m 1,..., m t, m t+1 ), (n 1,..., n t, n t+1 ) N t+1 0, we define the other weighted reverse lexicographic monomial order u in F q [X 1,..., X t, Z] such that (m 1,..., m t, m t+1 ) u (n 1,..., n t, n t+1 ) if a 1 m 1 + +a t m t + um t+1 > a 1 n 1 + +a t n t + un t+1, or a 1 m a t m t + um t+1 = a 1 n 1 + +a t n t + un t+1, and m 1 = n 1, m 2 = n 2,..., m i 1 = n i 1, m i < n i, for some 1 i t+1. As done in [8], the interpolation polynomial is the smallest nonzero polynomial with respect to u in the preimage of I r,m. Such a smallest element can be found from a Gröbner basis of the F q [x 1 ]- module I r,m,l in Theorem 8. To find such a Gröbner basis, Lee and O Sullivan proposed the following general purpose algorithm as [8, Algorithm G]. Their algorithm [8, Algorithm G] efficiently finds a Gröbner basis of submodules of F q [x 1 ] s for a special kind of generating set and monomial orders. Please refer to [1] for Gröbner bases for modules. Let e 1,..., e s be the standard basis of F q [x 1 ] s. Let u x, u 1,..., u s be positive integers. Define the monomial order in the F q [x 1 ]-module F q [x 1 ] s such that x n 1 1 e i LO x n 2 1 e j if n 1 u x + u i > n 2 u x + u j or n 1 u x + u i = n 2 u x + u j and i > j. For f = s i=1 f i(x 1 )e i F q [x 1 ] s, define ind( f )=max{i f i (x 1 ) 0}, where f i (x 1 ) denotes a univariate polynomial in x 1 i=0 8

9 over F q. Their algorithm [8, Algorithm G] efficiently computes a Gröbner basis with respect to LO of a module generated by g 1,..., g s F q [x 1 ] s such that ind(g i )=i. The computational complexity is also evaluated in [8, Proposition 16]. Letl be the maximum Z-degree of the interpolation polynomial in [5]. The set I r,m,l in Theorem 8 is an F q [x 1 ]-submodule of F q [x 1 ] a 1(l+1) with the module basis {y j Z k j=0,..., a 1 1, k=0,...,l}. Assumption 9 We assume that we have f L( Q) whose zero divisor ( f ) 0 = D. Observe that Assumption 9 is implied by [2, Assumption 2] and is weaker than [2, Assumption 2]. Let f be the ideal of L( Q) generated by f. By [10, Corollary 2.3] we havel( D+ Q)= f. By [10, Corollary 2.5] we havel( id+ Q)= f i. Example 10 This is continuation of Example 2. Let f = x We see that v Q ( f )=21 and that there exist 21 distinct F 8 -rational places P 1,..., P 21, such that f (P i ) = 0 for i = 1,..., 21 by straightforward computation. By setting D=P P 21 Assumption 9 is satisfied. We remark that we have v Q (x x 1)=24 but there exist only 23 F 8 -rational places P such that (x x 1)(P)=0, other than Q, and that (x x 1) does not satisfy Assumption 9. Without loss of generality we may assume existence of x L( Q) such that f F q [x ]. By changing the choice of x 1,..., x t if necessary, we may assume x 1 = x and f F q [x 1 ] without loss of generality, while it is better to make v Q (x 1 ) as small as possible in order to reduce the computational complexity. Under the assumption f F q [x 1 ], f i y j satisfies the required condition forη i, j in Theorem 8. By naming y j Z k as e 1+ j+ku, the generators in Theorem 8 satisfy the assumption in [8, Algorithm G]. In the following, we assign weight iv Q (x 1 ) v Q (y j )+ku to the module element x i 1 y jz k. With this assignment of weights, the monomial order LO is the restriction of u to the F q [x 1 ]-submodule ofl( Q)[Z] generated by {y j Z k j=0,..., a 1 1, k=0,...,l}. We can efficiently compute a Gröbner basis of the F q [x 1 ]-module I r,m,l by [8, Algorithm G]. After that we find the interpolation polynomial required in the list decoding algorithm by Guruswami and Sudan [5] as the minimal element with respect to LO in the computed Gröbner basis. Proposition 11 Suppose that we use [8, Algorithm G] to find the Gröbner basis of I r,m,l with respect to LO. Under Assumption 9, the number of multiplications 9

10 in [8, Algorithm G] with the generators in Theorem 8 is at most [max j { v Q (y j )}+m(n+2g 1)+u(l m)] 2 a 1 1 a 1 (l+1) i=1 i 2. (4) Proof. What we shall do in this proof is substitution of variables in the general complexity formula in Lee and O Sullivan [8] by specific values. The number of generators is a 1 (l+1), which is denoted by m in [8, Proposition 16]. We have v Q ( f ) n+g and v Q (h r ) n+2g 1. We can assume u n+2g 1. Thus, the maximum weight of the generators is upper bounded by max{ v Q (y j )}+m(n+2g 1)+u(l m). j By [8, Proof of Proposition 16], the number of multiplications is upper bounded by Eq. (4). Example 12 Consider the [21, 10] code C 12 over the Klein quartic considered in Examples 1, 2 and 10. Its Goppa bound is n u=21 12=9. The Guruswami and Sudan [5] algorithm can correct 5 errors with m=4 andl=7. For this code, only Guruswami and Sudan [5] can be used for the interpolation step. It requires us to solve a system of 21 ( ) = 210 linear equations. Solving such a system needs roughly /3=3, 087, 000 multiplications in F 8. The value of Eq. (4) is given by [max j { v Q (y j )}+m(n+2g 1)+u(l m)] 2 a = [ (7 4)] 2 /3+ = / /6 = i=1 i 2 1 a 1 (l+1) We see that the proposed method can solve the interpolation step much faster than Guruswami and Sudan [5]. i=1 i 2 10

11 4 Concluding Remarks The interpolation step in Guruswami and Sudan [5] is computationally costly and many researchers proposed faster interpolation methods, as summarized by Beelen and Brander [2, Figure 1]. However, those researches assumed either Hermitian curves, e.g. Lee and O Sullivan [8], Sakata [16] or C ab curves e.g. [2, 9]. Our argument used no assumption until Assumption 9 that seems indispensable with application of Algorithm G in Lee and O Sullivan [8]. The Klein quartic is the well-known family for constructing AG codes, but there seemed no faster alternative to the original interpolation step by Guruswami and Sudan [5] before our proposal. In Example 12 we demonstrated that the proposed interpolation procedure is much faster than the original [5] for codes on the Klein quartic. References [1] W. W. Adams and P. Loustaunau. An Introduction to Gröbner Bases, volume 3 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, [2] P. Beelen and K. Brander. Efficient list decoding of a class of algebraicgeometry codes. Adv. Math. Commun., 4(4): , [3] D. Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry, volume 150 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, [4] O. Geil and R. Pellikaan. On the structure of order domains. Finite Fields Appl., 8(3): , July [5] V. Guruswami and M. Sudan. Improved decoding of Reed-Solomon and algebraic-geometry codes. IEEE Trans. Inform. Theory, 45(4): , Sept [6] T. Høholdt and R. Pellikaan. On the decoding of algebraic-geometric codes. IEEE Trans. Inform. Theory, 41(6): , Nov [7] R. F. Lax. Generic interpolation polynomial for list decoding. Finite Fields Appl., 18(1): , Jan doi: /j.ffa [8] K. Lee and M. E. O Sullivan. List decoding of Hermitian codes using Gröbner bases. J. Symbolic Comput., 44(12): , Dec arxiv:cs/ [9] J. B. Little. List decoding for AG codes using Gröbner bases. Presented 11

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