General error locator polynomials for nth-root codes

General error locator polynomials for nth-root codes Marta Giorgetti 1 and Massimiliano Sala 2 1 Department of Mathematics, University of Milano, Italy 2 Boole Centre for Research in Informatics, UCC Cork, Ireland Abstract. All interesting linear codes (i.e., with d 2) form a class, called nth-root codes. We investigate the decoding for an interesting subclass, proving the existence of general error locator polynomials. 1 Introduction We introduce a class of linear codes called n-th root codes, that essentially includes all linear codes (as soon as their minimum distance is greater than or equal to two). Those codes are defined by means of their parity-check matrix, the expression of which generalizes the one for cyclic codes. In [8] the notation of general error locator polynomials for correctable linear codes was introduced. These are multivariable polynomials such that their specializations (to a correctable syndrome s) give the error locations (corresponding to s). We exhibit a subclass of linear codes for which general error locator polynomials do exist. To do so, we construct an ideal involving the polynomials defining the parity-check matrix of a (proper maximal zerofree) n-th root code. We investigate properties of this ideal and show that its totally reduced Gröbner basis contains a (unique) general error locator polynomial. In the same spirit, we show the existence of general locator polynomials of type ν - handling errors and erasures - for such codes. 2 Preliminaries We denote by F q the finite field with q elements, where q is a power of a prime, and by n a natural number such that q and n are relatively prime. Let k, N N such that 1 k N n + 1. We refer to the vector space of dimension N over F q as to (F q ) N. The zeros of polynomial x n 1, which are called n-th roots of unity, lie in an extension field F q m and in no smaller field. We denote the set of all these roots by R n. From now on, q,n, k, N and m are understood. All the following statements and definitions can be found in [5] and [4], unless otherwise stated.

2.1 General error locator polynomial Let C be an [N, k, d] code over F q, t its correction capability and H a paritycheck matrix over F q m. Let d 3. The syndromes lie in (F q m) N k and form a vector space of dimension (N k) over F q. Let α be a primitive N-th root of unity in F q m, so that n = N. Let r = N k. Definition 1 ([8]). Let L C be a polynomial in F q [X, z], where X = (x 1,..., x r ). Then L C is a general error locator polynomial of C if 1. L C (X, z) = z t + a t 1 z t 1 + + a 0, with a j F q [X], 0 j t 1, that is, L C is a monic polynomial with degree t with respect to the variable z and its coefficients are in F q [X]; 2. given a syndrome s = (s 1,... s r ) (F q m) N k, corresponding to a vector error of weight µ t and error locations {k 1,..., k µ }, if we evaluate the X variables in s, then the roots of L C (s, z) are {α k1,..., α kµ, 0,..., 0}. }{{} t µ Definition 2 ([8]). Let L be a polynomial in F q [X, W, z], X = (x 1,..., x r ) and W = (w ν,..., w 1 ), where ν 1 is the number of erasures that occurred. Then L is a general error locator polynomial of type ν of C if 1. L(X, W,z) = z τ + a τ 1 z τ 1 + + a 0, with a j F q [X, W ], for any 0 j τ 1, that is, L is a monic polynomial with degree τ in the variable z and coefficients in F q [X, W ]; 2. for any syndrome s = (s 1,..., s r ) and any erasure location vector w= (w 1,..., w ν ), corresponding to an error of weight µ τ and error locations {k 1,..., k µ }, if we evaluate the X variables in s and the W variables in w, then the roots of L(s, w, z) are {α k1,..., α kµ, 0,..., 0}. }{{} τ µ If such L exists for a given code C, then we name the polynomial L ν C. We denote by L 0 C polynomial L C. For a code C, the possession of a general locator polynomial L ν C of type ν for all 0 ν < d might be a stronger condition than the possession of a general error locator polynomial L C, but in [8] the authors prove that any cyclic code admits a general locator polynomial of type ν, for 0 ν < d. 2.2 Definition and first properties of nth-root codes Definition 3. Let L R n {0}, L = {l 1,..., l N } and P= {g 1 (x),..., g r (x)} in F q m[x] such that i = 1,..., N there is at least one j = 1,..., r such that g j (l i ) 0. We denote by C = Ω(q, n, q m, L, P) the linear code defined over F q having g 1 (l 1 )... g 1 (l N ) g 1 (L) g 2 (l 1 )... g 2 (l N ) g 2 (L) H =.. = g r (l 1 )... g r (l N ). g r (L)

as its parity-check matrix. We say that C is an nth-root code. Remark 1. Code C = Ω(q, n, q m, L, P) is linear over F q, its length is N = L and its distance d is greater than or equal to 2, because there are no columns in H composed only of zeros. If 0 L we assume l N = 0 (any re-ordering of L gives an equivalent code). Definition 4. Let C = Ω(q, n, q m, L, P) be an nth-root code. If L = R n \ L =, we say that C is maximal. If P F q [x], we say that C is proper. If 0 / L, we say that C is zerofree, non-zerofree otherwise. Since any function from F q m to itself can be expressed as a polynomial, we can accept in P also rational functions of type f/g, f, g F q m, such that g( x) 0 for any x F q m. We do so from now on, without further comments. Example 1. Let q = 2, n = 7, q m = 8, L = F 2 3 = β {0}, the minimal polynomial of β is z 3 1 + z + 1, and P= {g 1 (x) = x 2 +x+1, g x 2(x) = x 2 +x+1 }. The seven 7th roots of unity are all the elements of F 8, R 7 = F 8. The nth-root code C = Ω(2, 7, 8, F 8, {g 1, g 2 }) is non-zerofree (0 L), maximal and it is easy to see that C is an [8,2,5] code. Remark 2. In order to define the same nth-root code, it is possible to use different n. For example, to define an nth-root code with length N = 5, we can use the five 5th roots of unity or five 7th roots of unity. Proposition 1. Let C be a linear code over F q of length N and d 2. Then C is an nth-root code for any n N 1 such that (n, q) = 1. In particular: 1. if n = N, then C can be maximal zerofree, 2. if n = N 1, then C is maximal non-zerofree. Corollary 1. Let C be a linear code. C is an nth-root code if and only if d 2. 3 General error locator polynomial 3.1 Application of the Gianni-Kalkbrener Theorem Let K be a (not necessarily finite) field. Assume G is a Gröbner basis for a 0 dimensional ideal J K[S, A, T ], S = (s 1,..., s H ), A = (a 1,..., a L ), T = (t 1,..., t M ) w.r.t. a order with S < A < T and with the A variables lexicographically ordered by a 1 > a 2 >... > a L. Then the elements of set G (K[S, A] \ K[S]) can be collected into blocks {G i } 1 i L : G 1 = {g 1,1 (S, a L,..., a 1 ),..., g 1,l1 (S, a L,..., a 1 )}, G 2 = {g 2,1 (S, a L,..., a 2 ),..., g 2,l2 (S, a L,..., a 2 )},. G L = {g L,1 (S, a L ),..., g L,lL (S, a L )},

in such a way that: for any i, G i K[S, a L,..., a i+1 ][a i ] \ K[S, a L,..., a i+1 ], the ideal generated by j>i G j is actually the i-th elimination ideal J i, J i = J K[S, a L,..., a i ]. The Gianni-Kalkbrener Theorem [3] ensures that G i for 1 i L. Clearly any G i, 1 i L, can be decomposed into blocks of polynomials according to their degree with respect to the variable a i : G i = i δ=1 G iδ, but some G iδ could be empty. In this way, if g G iδ, we have: g K[S, a L,..., a i+1 ][a i ] \ K[S, a L,..., a i+1 ], deg ai (g) = δ, i.e. g = ba δ i + and b = Lp(g) K[S, a L,..., a i+1 ]. Let N iδ be the number of elements of G iδ. We name the elements of set G iδ = {g iδj, 1 j N iδ } according to their order: h < j Lt(g iδh ) < Lt(g iδj ). Remark 3. We can summarize our description. Given any two polynomials g ldh G ld and g iδj G iδ, then l > i or g ldh < g iδj Lt(g ldh ) < Lt(g iδj ) l = i, D < δ or l = i, D = δ, h < j Since J is 0 dimensional, we can clearly decompose the variety of its elimination ideals as follows. Let J S = J K[S], J S {al } = J K[S, a L ],..., J S {al,...,a 1} = J K[S, a L,..., a 1 ] = J K[S, A]. We have: 1) V(J S ) = λ(l) j=1 ΣL j, with Σ L j = {(s 1,..., s N ) V(J S ) there are exactly j values {ā (1) L,..., ā(j) L }, s.t.(s 1,..., s N, ā (i) L ) V(J S {a L }), 1 i j}; 2) V(J S {al }) = λ(l 1) Σ L 1 j j=1 Σ L 1 j,with = {(s 1,..., s N, a L ) V(J S {al }) there are exactly j values {ā (1) L 1,..., ā(j) L 1 }, s.t.(s 1,..., s N, a L, ā (i) L 1 ) V(J S {a L,a L 1 }), 1 i j}; 3) V(J S {al,...,a h }) = λ(h 1) Σ h 1 j j=1 Σ h 1 j, 2 h L with = {(s 1,..., s N, a L,..., a h ) V(J S {al,...,a h }) exactly j values {ā (1) h 1,..., ā(j) h 1 }, s.t.(s 1,..., s N, a L,..., a h, ā (i) h 1 ) V(J S {a L,...,a h+1 }), 1 i j}; (1)

Note that, for a general 0-dimensional ideal J, nothing can be said about λ(h), except that λ(h) 1 for any 2 h L. We now introduce a class of ideals which are very useful in our context. Definition 5. With the above notation we say that J is stratified if: 1. λ(h) = h, 1 h L and 2. h j, 1 h L, 1 j h. Example 2. Let S = {s 1 }, A = {a 1, a 2 } (so that L = 2) and T = {t 1 } such that S < A < T and a 1 > a 2. Let K = C and J be the ideal in C[s 1, a 1, a 2, t 1 ] generated by: {s 2 1 s 1, a 2 3, a 1 s 1 2s 1, a 2 1 + a 1 s 1 3a 1 2s 1 + 2, t 1 }. The variety of J is V(J) = {(0, 1, 3, 0), (0, 2, 3, 0), (1, 2, 3, 0)}. Let J S = J C[S] = s 2 1 s 1, then V(J S ) = J S = λ(l) L j=1 j = 2 λ(2) j=1 j = {0, 1}. Clearly {1} = 2 1, which means λ(2) = 2 satisfying condition (1) in Definition 5, for h = 1, 2. Variety V(J S {a2}) = λ(l 1) λ(1) j=1 j = {(0, 1), (0, 2), (1, 2)}. Clearly {(0, 1), (0, 2), (1, 2)} = 1 1 which means λ(l 1) = λ(1) = 1 satisfying condition (1) and all i j, i, j = 1, 2, are not empty, so that ideal J is stratified. See Figure 1 (A) and (B). and {0} = 2 3 Theorem 1. Let J be a radical stratified ideal, then for 1 i L, G i = i δ=1g iδ, with G iδ, 1 δ i and 1 i L. Moreover 1 i L, G ii = {g ii1 }, i.e. only one polynomial exists in G i with degree i w.r.t. a i ; 1 i L, Lp(g ii1 ) = 1, Lt(g ii1 ) = a i i. 3.2 Ideals for the decoding of nth-root codes Definition 6. Let C = Ω(q, n, q m, L, P) be a zerofree maximal nth-root code, with correction capability t. We denote by J C,t the ideal J C,t F q m[x 1,..., x r, z t,..., z 1, y 1,..., y t ], { t } { } J C,t = h=1 y hg s (z h ) x s, y q 1 j 1, 1 s r 1 j t {z i z j p(z i, z j )} i j, 1 i,j t, { zj n z } (2) j 1 j t where p(x, y) = n 1 h=0 xh y n 1 h. We denote by G C,t the totaly reduced Gröbner basis of J C,t w.r.t. >. Note that variables x 1,..., x r represent correctable syndromes, z 1,..., z t error locations and y 1,..., y t error values.

5 a 2 (A) a 1 4 5 4 (B) 3 3 2 2 1 1 0 0 1 2 3 4 5 s 1 0 0 (1,0) (2,0) (2,1) (a,s ) 2 1 Fig. 1. Varieties in a stratified case. Lemma 1. Ideal J C,t is radical and stratified. Applying Theorem 1 to J C,t (and Lemma 1), we have the following proposition. Proposition 2. In Gröbner basis G C,t there exists a unique polynomial of type g = z t t + a t 1 z t 1 t We now state the main result of this paper. +... + a 0, a i F q m[x]. Theorem 2. If code C is a proper maximal zerofree nth-root code with correction capability t, then C possesses a general error locator polynomial. Since cyclic codes are proper maximal zerofree nth-root codes we obtain, as a special case of Theorem 2, that cyclic codes have general error locator polynomials (Theorem 6.9 in [8]). From now on we often shorten general error locator polynomial to OS polynomial. In the next two examples we show two methods to compute OS polynomials. The former is suggested by Proposition 2. In the latter we assume we know that a general error locator polynomial exists for the code and hence we apply directly Definition 1. Example 3. Let G and H be following binary matrices ( ) 1 1 1 0 0 G = H = 1 0 1 0 1 0 1 1 0 1. 0 0 1 1 1 0 0 0 1 1

Let C be the [5, 2, 3] linear code over F 2 with G as a generator matrix and H as a parity-check matrix. Note t = 1. Let γ be a primitive element of F 16 with minimal polynomial z 4 + z + 1. Then C is the zerofree maximal nth-root code Ω(2, 5, 2 4, R 5, P), where P = { g 1 (x) = γ 4 x 4 + γ 8 x 3 + γ 2 x 2 + γx + 1, g 2 (x) = γ 10 x 4 + γ 5 x 3 + γ 5 x 2 + γ 10 x + 1, g 3 (x) = γ 11 x 4 + γ 7 x 3 + γ 13 x 2 + γ 14 x}. We construct ideal J C,t F 16 [x 1, x 2, x 3, z 1 ] = F 16 [X, Z], as follows: J C,1 = {g h (z 1 ) x h } 1 h 3, z n 1 z 1. If we calculate Gröbner basis G C,t = G X G X,z1 induced by x 1 < x 2 < x 3 < z 1, we obtain: w.r.t. the lexicographical order G X = {x 2 3 + x 3, x 2 2 + x 2, x 1 x 3 + x 2 x 3, x 1 x 2 + x 1 + x 2 x 3 + x 2 + x 3 + 1, x 2 1 + x 1 } and G X,z1 = {g 111 = z 1 + (γ 2 +γ)x 1 + (γ 3 +γ)x 2 x 3 + γx 2 + x 3 + (γ 3 +γ 2 +γ)}. In G X,z1 there is only one polynomial in z 1 of degree 1, as we expected, g 111, and it must be an OS polynomial for C thanks to Fact 2. Example 4. Let C be the code in Example 3. Another way to compute an OS polynomial is to see code C with parity-check matrix H = (γ 6, γ 2, γ 3, γ 14, 1), so that C = Ω(2, 5, 2 4, R 5, P ), where P = {γ 12 x 4 + γ 11 x 3 + x 2 + γ 14 x + γ 3 }. If we calculate the Gröbner basis G w.r.t. the lexicographical order induced by x 1 < z 1, its elements are: G x 1 = x 5 1 + (γ 3 )x 4 1 + (γ 3 + γ)x 2 1 + γ 2 x 1 + (γ 2 + γ + 1), G x 1,z 1 = z 1 + x 3 1. There is only one polynomial in z 1 of degree 1, as we expected, and it is another OS polynomial for C. Example 5. Another way to compute OS polynomials for a code is to suppose that those polynomials exist. Let C be the code studied in Example 3. We assume that its parity-check matrix is a row, H = (e 1, e 2, e 3, e 4, e 5 ). We search for an OS polynomial z + f(x) (the degree t of z is 1). It must satisfy the following conditions: f(e i ) = α i, 1 i 5, and f(0) = 0. Polynomial f(x) has degree at most 5 with coefficients b i in F 2, so that we can write f(x) = b 5 x 5 + b 4 x 4 + b 3 x 3 + b 2 x 2 + b 1 x (f(0) = 0 b 0 = 0). We compute a Gröbner basis of ideal J F 16 [b 1, b 2, b 3, b 4, b 5, e 2, e 3, e 5 ] given by J = e 1 + e 2 + e 3, e 3 + e 4 + e 5, {e 15 i + 1} 1 i 5, {b 2 i + b i} 1 i 5, f(e 1 ) + γ 3, f(e 2 ) + γ 6, f(e 3 ) + γ 9, f(e 4 ) + γ 12, f(e 5 ) + γ 15

where relations e 1 = e 2 + e 3, e 4 = e 3 + e 5 follow from matrix G. We obtain e 1 = γ 6, e 2 = γ 2, e 3 = γ 3, e 4 = γ 14, e 5 = 1, so that the parity-check matrix is H = (γ 6, γ 2, γ 3, γ 14, 1) and the OS polynomial is f(x) = x 3. We note that it is the same as in Example 4. Remark 4. The previous example is interesting because we have simultaneously computed for C an nth-root presentation and a general error locator polynomial. The nice shape of the general error locator polynomial reveals an unexpected structure in this code. If the approach presented in Example 5 fails for a code C, that is, if V(J) =, then it means that C does not possess an OS polynomial for any nth-root presentation such that H is composed of one row. However, it could be that C possesses an OS polynomial for H with up to N k rows. We think that it is obvious how this may be checked with a similar commutative algebra approach, and so we do not detail it. Example 6. Consider the nth-root code of Example 1, shortened in position 0. It is a classical Goppa code with g(x) = x 2 + x + 1 and L = F 8. An OS polynomial for this code is L = z 2 2 + z 2 (x 5 1x 2 2 + x 5 1 + x 3 1x 2 2 + x 3 1 + x 2 1x 2 2 + x 2 1x 2 + x 1 x 5 2 + x 1 x 4 2 + x 1 x 3 2+ x 1 x 2 2 + x 1 x 2 + x 1 + x 7 2 + x 4 2 + x 3 2 + x 2 2 + 1) + x 5 1x 2 2 + x 5 1x 2 + x 5 1 + x 4 1x 2 2+ x 3 1x 3 2 + x 2 1x 2 + x 2 1 + x 1 x 6 2 + x 1 x 2 + x 1 + x 7 2 + x 6 2. 3.3 Extended syndrome variety We extend previous results to the case when there are also erasures. Let τ be a natural number corresponding to the number of errors, µ be a natural number corresponding to the number of erasures and such that 2τ + µ < d. We have to find solutions of equations of type: s j + τ a l g j (α k l ) + l=1 ν l=1 c lg j (α h l ), j = 1,..., r (3) where {k l }, {a l } and {c l} are unknown and { s j }, {h l} are known. We introduce variables W = (w ν,..., w 1 ) and U = (u 1,..., u ν ), where the {w h } stand for erasure locations (α h l ) and the {uh } stand for erasure values c l (h = 1,..., ν). When the word v(x) is received, the number ν of erasures and their positions {w h } are known. We rewrite equations (3) in terms of X, Y, Z, W and U, where the {x j } stand for the syndromes (j = 1,..., r), as: J C,τ,ν = { τ l=1 y lg j (z l ) + ν l u lg j (w l) x j } j=1,...,r,, {z n+1 i z i } i=1,...,τ, {y q 1 i 1} i=1,...,τ, {u q h u h} h=1,...,ν, {wh n 1} h=1,...,ν, {x qm j x j } j=1,...,r, {p(w h, w k )} h k,h,k=1,...,ν, {z i p(z i, w h )} i=1,...,τ,h=1,...,ν, {z i z j p(z i, z j )} i j,i,j=1,...,τ.

We observe that: - polynomials τ l=1 y lg j (z l ) + ν l u lg j (w l) x j characterize the nth-root code; - polynomials z n+1 i z i ensure that z i are nth-roots of unity or 0; - polynomials y q 1 i 1, wh n 1, uq h u h ensure that y i, w h F q and u h F q ; - polynomials z i p(z i, w h ) ensure that an error cannot occur in a position corresponding to an erasure; - polynomials p(w h, w k ) ensure that any two erasure locations are distinct; - polynomials z i z j p(z i, z j ) ensure that any two error locations are distinct. Ideal J C,τ,ν depends only on code C and on ν. Proposition 3. In Gröbner basis G C,τ,ν there is a unique polynomial of type g = z τ τ + a τ 1 z τ 1 +... + a 0, a i F q m[x, W ]. Theorem 3. If code C is a proper maximal zerofree nth-root code, then C possesses general error locator polynomials of type ν, for any ν 0. Example 7. Let C be the shortened code obtained from code C presented in Example 1. Code C is a [7, 1, 6] linear code, so that τ (errors) and µ (erasures) satisfy relation τ + µ < 6. If τ = 1, µ = 2, the syndrome ideal is J = {g 1 (z 1 ) + u 1 g 1 (w 1 ) + u 2 g(w 2 ) + x 1, g 2 (z 1 ) + u 1 g 2 (w 1 ) + u 2 g 2 (w 2 ) + x 2, z 8 1 z 1, w 7 1 1, w 7 2 1, x 8 1 x 1, x 8 2 + x 2, u 2 1 + u 1, u 2 2 + u 2, z 1 p(z 1, w 1 ), z 1 p(z 1, w 2 ), p(w 1, w 2 )} and in the reduced Gröbner basis there is only one polynomial having z 1 as leading term (see Appendix of [5]). 4 Conclusions and further research Linear codes are traditionally specified starting from a parity-check matrix H. In particular, cyclic codes are such that the entries of H consist of the evaluation of univariate monomials on all the n-th roots of unity. Our approach is to specify any linear code (with d 2) as a code such that the entries of H consist of the evaluation of generic (univariate) polynomials on all the n-th roots of unity. In this sense, we say that linear codes are a generalization of cyclic codes. This point of view allows to extend to linear codes some computational algebra techniques and some argument, that have been previously applied to cyclic codes ([6],[7],[8]). This translates in new tools, but also in new challenges. In particular, we can identify a new decoding algorithm for a (potentially very large) sub-class, via the general error locator polynomial. The problem of decoding linear codes is NP-hard ([1], [2]), but if a linear code admits a sparse general error locator polynomial (or such a polynomial with a sparse representation), then it can be decoded very fast ([9]). We have provided an explicit example when the locator polynomial is very small, given a certain nth-root presentation, and large when given another.

Acknowledgments The first author would like to thank the second author (her supervisor). The authors would like to thank the following people for their comments and suggestions: J. Abbot, M. Bardet, F. Caruso, F. Dalla Volta, J. C. Faugere, P. Fitzpatrick, T. Mora, E. Orsini, M. Pellegrini, L. Perret, I. Simonetti, C. Traverso. We have run our computer simulations using the software package Singular (http://www.singular.uni-kl.de) at the computational centre MEDICIS (http://medicis.polytechnique.fr). This work has been partially supported by the STMicroelectronics contract Complexity issues in algebraic Coding Theory and Cryptography. References 1. A. Barg, Complexity issues in coding theory, Handbook of coding theory, Vol. I, II, p. 649 754, North-Holland, Amsterdam, 1998. 2. A. Barg, E. Krouk, H. C. A. van Tilborg, On the complexity of minimum distance decoding of long linear codes, IEEE Trans. Inform. Theory, vol. 45, 1999, no. 5, p. 1392 1405. 3. M. Caboara, T. Mora, The Chen-Reed-Helleseth-Truong decoding algorithm and the Gianni-Kalkbrenner Groebner shape theorem, Applicable Algebra in Engineering, Communication and Computing, vol. 13, 2002, no. 3, p. 209 232. 4. M. Giorgetti, On some algebraic interpretation of classical codes, PhD Thesis, University of Milan, 2007. 5. M. Giorgetti, M. Sala, A commutative algebra approach to linear codes, UCC- BCRI preprint, www.bcri.ucc.ie, 58, 2006. 6. M. Sala, Groebner basis techniques to compute weight distributions of shortened cyclic codes, Journal of Algebra and Its Applications, vol. 6, no. 2, 2007 7. M. Sala, Groebner bases and distance of cyclic codes, Applicable Algebra in Engineering, Communication and Computing, vol. 13, 2002, no. 2, p. 137 162. 8. E. Orsini, M. Sala, Correcting errors and erasures via the syndrome variety, Journal of Pure and Applied Algebra, vol. 200, 2005, no. 1-2, p. 191 226. 9. E. Orsini, M. Sala, General error locator polynomials for binary cyclic codes with t 2 and n < 63, IEEE Trans. Inform. Theory, vol. 53, 2007, no. 3, p. 1095 1107.