Partial Permutation Decoding for abelian codes

Transcription

1 1 Partial Permutation Decoding for abelian codes José Joaquín Bernal Juan Jacobo Simón Abstract In [3], we introduced a technique to construct an information set for every semisimple abelian code over an arbitrary field, solely in terms of its defining set. In this paper we apply the geometrical properties of those information sets to obtain sufficient conditions for a t-error correcting abelian code to have a b-pd-set for every b t. These conditions are simply given in terms of the structure of the defining set of the code. I. INTRODUCTION Permutation decoding was introduced by F. J. MacWilliams in [14] it is fully described in [9] [15]. For a fixed information set of a given linear code, this technique uses a special set of permutation automorphisms of the code called PD-set. The idea of permutation decoding is to apply the elements of the PD-set to the received vector until the errors are moved out of the fixed information set. But, how can we determine when all errors have been moved out of the information positions? Given a t-error correcting code with a fixed information set parity check matrix in stard form, it is proved (see, for example, [9, Theorem 8.1] that a received vector has syndrome with weight less than or equal to t if only if all its information symbols are correct. Finding adequate information sets PD-sets is not trival. Many authors have studied families of codes for which it is possible to develop methods to find PD-sets with respect to certain types of information sets. in all cases the techniques used depend on family which is being studied. We are interested in the family of abelian codes. Some important families of codes are abelian, for instance: cyclic codes, Reed-Muller codes, extended Reed-Solomon codes others. The existence of PD-sets relies on the information set considered as reference. In fact, it may happen that for an error correcting code the selection of the information set causes the non-existence of a PD-set. Hence the importance of methods algorithms to construct information sets. In the case of cyclic codes, McWilliams uses the well-known fact that for a cyclic code of dimension k, any selection of k consecutive positions defines an information set. In [10], H. Imai gave a method to obtain information sets for binary two dimensional cyclic (TDC codes of odd area. Later, S. Sakata [16] gave an alternative method for the same purpose. Imai s algorithm relies on the structure of the roots of the code, while the algorithm of Sakata is somehow based on the division algorithm for polynomials. Up to our knowledge, these are the sole techniques for TDC codes. Following the ideas in the two papers mentioned above, H. Chabanne [7] gave a method to calculate syndromes by using the division algorithm for polymonials in several variables Groebner basis; by using it he generalized the McWilliam s permutation decoding procedure. The techniques used by Chabanne involve a generalization of the information sets obtained by Sakata to binary abelian codes. In [3] we presented a method for constructing information sets valid for every semisimple abelian code, not necessarily binary. It is based on the computation of the cardinalities of certain cyclotomic cosets on different extensions of the ground field it generalizes Imai s method in the case of TDC codes. Such cosets are completely determined by the structure of the defining set of the code. This technique allows us to design codes with suitable information sets in order to use permutation decoding (see [4]. In this paper we find sufficient conditions for an abelian code, viewed as an ideal of a multivariate polynomial quotient ring, to have a PD-set contained in the translations associated to each variable. Moreover, the goal of this paper is that such conditions may be written solely in terms of the q-cyclotomic structure of the defining set of abelian codes (see below for all definitions. In Section II we review basic facts about abelian codes permutation decoding. In Section III we reproduce without proofs the construction of sets of check positions ( hence information sets given in [3]. In Section IV, we apply the results of the previous section to get sets of check positions for a code C its dual C. Then, as a first part of our main results, we study the relationship between them, which we denote by Γ(C Γ(C respectively. More precisely, we show that there exists a simple bijection from the complementary set Γ(C c to Γ(C (Theorem 14. Among other applications, this allows us to show that it is equivalent to use one or the other in order to determine a PD-set. We also show that the set Γ(C may be determined from the complement of the set of roots of C (Corollary 11. In Section V we include the second part of our An extended abstract of this paper, with the same title, appears in the proceedings of 2012 IEEE International Symposium on Information Theory (ISIT 2012, Cambridge, Mass. Departamento de Matemáticas, Universidad de Murcia, Murcia. Spain. josejoaquin.bernal@um.es, jsimon@um.es Partially supported by MINECO (Ministerio de Economía y Competitividad, (Fondo Europeo de Desarrollo Regional project MTM Fundación Séneca of Murcia.

2 2 main results, namely, we give sufficient conditions for a semisimple abelian code to have a partial PD-set in the set of those translations associated to each variable. These conditions make use of the previous results about the dual code (Theorems 21, 22, 23 Proposition 26. Finally, Section VI contains applications that show us how we may use the conditions obtained to design exhibit codes that improve the parameters of best known permutation decodable abelian codes for certain lengths. II. PRELIMINARIES Throughout F denotes the field with q elements where q is a power of a prime p. Let C be a linear code of dimension k, length l over the field F, that is, a subspace of F l with dimension k. We call the elements of C codewords. An information set for C is a set of positions {i 1,..., i k } {1,..., l} such that restricting the codewords to these positions we get the whole space F k. For every codeword the symbols in the positions corresponding to an information set are called information symbols the other l k positions are called check positions [13]. A generator matrix for C is a k l matrix G whose rows form a basis for C. We say that G is in stard form if it is of the form [I k A], where I k is the identity matrix of order k. We denote by C the dual code of C under the ordinary inner product, that is, C = {v F l u v = 0 for all u C}. A parity check matrix for C is a generator matrix for C. If G is a generator matrix in stard form, it is easy to check that H = [ A T I l k ] is a parity check matrix. In this case we say that H is also in stard form. As usual, for any codeword c C we denote its support by supp(c; that is, the set of its non-zero entries. We consider the parameter t = d 1 2, where d is the minimum distance of C, that measures the error-correction capability of C. Then we say that C is an [l, k] t-error-correcting code. We see the group of permutations on l symbols, S l, acting on F l via σ(c 1,..., c l = ( c σ 1 (1,..., c σ 1 (l with σ Sl. Then the permutation automorphism group of C is PAut(C = {σ S l σ(c = C}. Two linear codes C C are said to be permutation equivalent if there exists σ S l such that σ(c = C. It is easy to see that any linear code is permutation equivalent to a code which has a generator matrix in stard form. Now, we recall some basic facts about the family of abelian codes their permutation automorphisms (the reader may see [1] for details. A. Abelian codes An abelian code is an ideal of a group algebra FG, where G is an abelian group. It is well-known that a decomposition G C r1 C rn, with C ri the cyclic group of order r i, induces a canonical isomorphism of F-algebras from FG to F[X 1,..., X n ]/ X r1 1 1,..., Xrn n 1. We denote this quotient algebra by A(r 1,..., r n. So, we identify the codewords with polynomials P (X 1,..., X n such that every monomial satisfy that the degree of the indeterminate X i is in Z ri, the set of non negative integers less than r i. We write the elements P A(r 1,..., r n as P = P (X 1,..., X n = a j X j, where j = (j 1,..., j n Z r1 Z rn X j = X j1 1 Xjn n. We deal with abelian codes in the semisimple case, that is, we always assume that gcd(r i, q = 1 for every i = 1,..., n. Our construction makes use of the structure of roots of the ideals in A(r 1,..., r n ; so let us recall some basic facts about it. For a fixed primitive r i -th root of unity α i in some extension of F, i = 1,..., n, every abelian code C in A(r 1,..., r n is totally determined by its root set, The defining set of C with respect to α = {α 1,..., α n } is Z(C = {(α a1 1,..., αan n P (α a1 1,..., αan n = 0 for all P (X 1,..., X n C}. D α (C = {(a 1,..., a n Z r1 Z rn (α a1 1,..., αan n Z(C}. Given an abelian code C A(r 1,..., r n with defining set D α (C if one chooses different primitive roots of unity, say β = {β 1,..., β n }, then the set D β (C detemines a new code, say C, which is permutation equivalent to C. So, for the sake of brevity, we refer to abelian codes without any mention to the primitive roots that we are using as reference, we denote the defining set of C by D(C. Recall that, for γ N, the q γ -cyclotomic coset of an integer a modulo r is the set C (q γ,r(a = { a q γ i i N } Z r. We extend the concept of q-cyclotomic coset of an integer to several components.

3 3 Definition 1. Given an element (a 1,..., a n Z r1 Z rn, we define its q-orbit modulo (r 1,..., r n as Q(a 1,..., a n = {( a 1 q i,..., a n q i i N } Z r1 Z rn. It is easy to see that for every abelian code C A(r 1,..., r n, D (C is closed under multiplication by q in Z r1 Z rn, then D(C is necessarily a disjoint union of q-orbits modulo (r 1,..., r n. Conversely, every union of q-orbits modulo (r 1,..., r n defines an abelian code in A(r 1,..., r n. For the sake of simplicity we only write q-orbit, the tuple of integers will be clear from the context. The structure of q-orbits of the defining set is the essential ingredient for our algorithm of construction of information sets, which will be described in Section III. B. Permutation decoding This decoding algorithm was introduced by F. J. MacWilliams in [14]. The method is described fully in [15] [9]. For a fixed information set of a given linear code C, this technique uses a special set of permutation automorphisms of the code called PD-set. Definition 2. Let C be an [l, k] t-error-correcting code. Let I be an information set for C. For s t a s-pd-set for C I is a subset P PAut(C such that every set of s coordinate positions is moved out of I by at least one element of P. In case s = t, we say that P is a PD-set (see [12], [14]. Given a t-error-correcting code with a PD-set with respect to some information set, the idea of permutation decoding is to apply the elements of the PD-set to the received vector until the errors are moved out of the fixed information set. The following theorem shows how to check that the information symbols of a vector with weight less or equal than t are correct. We denote the Hamming weight of a vector v F l by wt(v. Theorem 3 ([9], Theorem 8.1. Let C be an [l, k] t-error-correcting code with parity check matrix H in stard form. Let r = c + e be a vector, where c C wt(e t. Then the information symbols in r are correct if only if wt ( Hr T t. Once we have found a PD-set P PAut(C for the given code C with respect to the information set I, the algorithm of permutation decoding is as follows: take a parity check matrix H for C in stard form. Suppose that we receive a vector r = c + e, where c C e represents the error vector satisfies that wt(e t. Then we calculate the syndromes H (τ(r T, with τ P, until we obtain a vector H (τ 0 (r T with weight less than or equal to t. By the previous theorem, the information symbols of the permuted vector τ(r are correct, so by using the parity check equations we get the redundancy symbols then we can construct a codeword c C. Finally, we decode to τ 1 (c = c. In general to find t-pd-sets for a given t-error correcting code is not at all an easy problem. It depends on the chosen information set. Moreover, it is clear that the algorithm is more efficient when the PD-set is small. ( We denote the permutation group on Z r1 Z rn by S r1 r n we consider it acting on A(r 1,..., r n via τ j a jx j = j a jx τ(j. From this point of view the permutation automorphism group of an abelian code C in A(r 1,..., r n may be described as PAut(C = {τ S r1 r n τ(c = C}. Let T j be the transformation from A(r 1,..., r n into itself, given by T j (P = X j P, for j = 1,..., n. Then it is clear that T j induces a permutation in S r1 r n, which we also denote by T j, via T j (i 1,..., i n = (i 1,..., i j + 1,..., i n. Then {T j } n j=1 may be viewed as a subgroup of permutation automorphisms for every abelian code in A(r 1,..., r n. We shall look for PD-sets contained in the subgroup {T j } n j=1. III. INFORMATION SETS IN ABELIAN CODES In this section we describe the method for the construction of sets of check positions for abelian codes (not necessarily binary given in [4]. It depends solely on the defining set of the code. The reader may see the mentioned paper for details. Let us consider the algebra A(r 1,..., r n under the assumptions gcd(r i, q = 1, for all i = 1,..., n, n 2. Let D be a union of q-orbits modulo (r 1,..., r n (see Definition 1. For each i = 1,..., n let D i denotes the projection of the elements of D onto the first i-coordinates. Then, given e = (e 1,..., e j D j, with 1 j n, we define γ(e = Q(e where q = q, in case j = 1, q = q γ(e1,...,ej 1 otherwise. m (e = C(q,r j(e j, (1

4 4 As we have noted, in the semisimple case every defining set of an abelian code in A(r 1,..., r n is a union of q-orbits modulo (r 1,..., r n. Our construction is based on the computation of the parameters (1 on a special set of representatives of the q-orbits. In fact, the representatives must satisfy the conditions given by the following definition. Definition 4. Let D be a union of q-orbits modulo (r 1,..., r n fix an ordering X i1 < < X in. A set D of representatives of the q-orbits of D is called a restricted set of represantives, with respect to the fixed ordering, if for every e = (e 1,..., e n e = (e 1,..., e n in D one has that, for all j = 1,..., n, the equality Q(e i1,..., e ij = Q(e i 1,..., e i j implies that (e i1,..., e ij = (e i 1,..., e i j. One can prove that restricted sets of representatives of a union of q-orbits always exist. Moreover, the construction of the information set does not depend on the selection on the representatives (see [4]. However, different orderings on the indeterminates may yield different information sets. From now on we consider as default ordering the following one: X 1 < < X n. Now we describe our construction. Let C A(r 1,..., r n be an abelian code with defining set D(C. Let D(C be a restricted set of representatives of the q-orbits in D(C, with respect to the default ordering on the indeterminates. As before, for each 1 i n, we denote by D i (C D i (C the projection onto the first i-coordinates of D(C D(C respectively. Given e D i (C, let R(e = {a Z ri+1 (e, a D i+1 (C}, (2 where (e, a has the obvious meaning; that is, if e = (e 1,..., e i then (e, a = (e 1,..., e i, a. For the algorithm we need to calculate n families of sequences of natural numbers. For each e D n 1 (C, we define M(e = m (e, a (3 a R(e consider the set {M(e} e Dn 1(C. Then we denote the different values of the M(e s as follows, f [1] = max e D n 1(C {M(e} f [i] = max e D n 1(C {M(e M(e < f [i 1]}. So, we obtain the sequence f [1] > > f [s] > 0 = f [s + 1], (4 that is, we denote by f [s] the minimun value of the parameters M( we set f [s + 1] = 0 by convention. Note that M(e > 0, for all e D n 1 (C, by definition. For any value of n, this is the initial family of sequences it is always formed by a single sequence. Now, suppose that n 3. Then we continue as follows: Given 1 u s, we define for every e D n 2 (C Ω u (e = {a R(e M(e, a f [u]} µ u (e = a Ω u(e m(e, a. Observe that the set Ω u (e may eventually be the empty set. In this case, the corresponding value µ u (e will be zero. We define We order the previous parameters we get the sequence f [u, 1] = max e D n 2(C {µ u (e} f [u, i] = max e D n 2(C {µ u (e 0 < µ u (e < f [u, i 1]}. f [u, 1] > > f [u, s(u] > 0 = f [u, s(u + 1], where f [u, s(u] denotes the minimum value of the parameters µ u ( f [u, s(u + 1] = 0 by definition. So we obtain the second family of sequences {f [u, 1] > > f [u, s(u] > 0 = f [u, s(u + 1] u = 1,..., s}.

5 5 In order to describe how to define a family of sequences from the previous ones, suppose that we have constructed the j-th family, when n 1 > j 1. For the sake of brevity, in what follows we denote δ = n j + 2. where, for every i = 2,..., n, {f [u n,..., u δ, 1] > > f [u n,..., u δ, s(u n,..., u δ ] > > 0 = f [u n,..., u δ, s(u n,..., u δ + 1] (u n,..., u δ Υ j (C} Υ i (C = {(u n,..., u n i+2 1 u n s (5 1 u δ s(u n,..., u δ+1 for δ = n i + 2,..., n 1}. For each (u n,..., u n j+2 Υ j (C we take the corresponding sequence: f [u n,..., u δ, 1] > > f [u n,..., u δ, s(u n,..., u δ ] > > 0 = f [u n,..., u δ, s(u n,..., u δ + 1]. Let u {1,..., s(u n,..., u n j+2 }. Then, for every e D n j 1 (C we define Ω un,...,u δ,u(e = {a R(e µ un,...,u δ (e, a f [u n,..., u δ, u]} µ un,...,u δ,u(e = a Ω un,...,uδ,u(e m(e, a. By ordering the different values µ un,...,u δ,u(e, with e D n j 1 (C, we obtain f [u n,..., u δ, u, 1] > > f [u n,..., u δ, u, s(u n,..., u δ, u] > 0 = f [u n,..., u δ, u, s(u n,..., u δ, u + 1], where f [u n,..., u δ, u, s(u n,..., u δ, u + 1] = 0 by convention. Then the (j + 1-th family of sequences is {f [u n,..., u δ 1, 1] > > f [u n,..., u δ 1, s(u n,..., u δ 1 ] > > 0 = f [u n,..., u δ 1, s(u n,..., u δ 1 + 1] (u n,..., u δ 1 Υ j+1 (C}. We follow the previous process until we get n 1 families of sequences. Finally, by using all the previous ones, we define, for any value of n, the last family of sequences. For every (u n,..., u 2 Υ n (C we define m(e if n = 2, So the last family of sequences is The algorithm yields the following set g[u n,..., u 2 ] = e D 1 (C M(e f[u 2 ] e D 1 (C µun,...,u 3 (e f[un,...,u 2 ] m(e if n > 2. {g[u n,..., u 3, 1] < < g[u n,..., u 3, s(u n,..., u 3 ] < (7 < g[u n,..., u 3, s(u n,..., u 3 ] (u n,..., u 3 Υ n 1 (C}. Γ(C = {(i 1,..., i n Z r1 Z rn (8 there exists (u n,..., u 2 Υ n (C such that f [u n,..., u j + 1] i j < f [u n,..., u j ], for j = 2,..., n, 0 i 1 < g[u n,..., u 2 ]}. The following theorem, proved in [4], establishes that Γ(C is a set of check positions for C. Theorem 5 ([4]. Let C be an abelian code in A(r 1,..., r n. Assume that gcd(r i, q = 1, for every i = 1,..., n, n 2. Then Γ(C is a set of check positions for C. We conclude this section with an example which will be referred to later. (6

6 6 Example 6. Let q = 2, n = 2, r 1 = 7, r 2 = 5, consider the abelian code C with the following defining set with respect to certain roots of unity: D(C = {(0, 0, (1, 1, (2, 2, (4, 4, (1, 3, (2, 1, (4, 2, (1, 4, (2, 3, (4, 1, (1, 2, (2, 4, (4, 3 }. We choose D(C = {(0, 0, (1, 1} as a complete set of restricted representatives (see Definition 4. Then, by following (1 (3 we compute M(0 = 1, M(1 = 4 m(0 = 1, m(1 = 3. Using these values we may get the sequences f [1] = 4 > f [2] = 1 > f [3] = 0 g[1] = 3 < g[2] = 4 which correspond to (4 (6 respectively. We use them to produce the marks in the following picture. f [1] f [2]..... f... [3] = 0. g[1] g[2] 7 So, a set of check positions for C is (see (8 Γ(C = {(0, 0, (0, 1, (0, 2, (0, 3, (1, 0, (1, 1, (1, 2, (1, 3, (2, 0, (2, 1, (2, 2, (2, 3, (4, 0}. IV. DUAL CODE For any abelian code C in A (r 1,..., r n, we denote by C its dual code. By the previous section, we know that Γ ( C the complementary set Γ(C c n i=1 Z r i are information sets for C. In this section, we will see that these information sets may be identified; that is, we will prove that there is a permutation κ S r1 r n such that for any abelian code C, κ PAut(C, furthermore, κ (Γ(C = Γ ( C c. Using this, we will conclude that, in order to apply the permutation decoding algorithm, both sets are equivalent. This fact will be used in the following section. Now, to relate information sets of abelian codes their duals we need to determine both sets of check positions with some notational compatibility. For this reason, we are going to add to the construction of Γ(C some trivial values of the parameters used. As the reader will see, they do not imply any changes in the information sets obtained. We note that this variant in the definition of the set of check positions will be used exclusively in this section. So we begin by considering an abelian code C with defining set D(C with respect to certain primitive roots of unity. To avoid any confusion with the original definition of Γ(C we will make a change on the symbols used to determine it. The new set determined by the new parameters will be denoted by Γ(C. Later, we will see that they coincide. Now we choose a set of restricted representatives D of the q-orbits of the whole set D = Z r1... Z rn as in Definition 4. Then, in order to construct Γ(C, we take the set D(C = D D(C as our set of restricted representatives of the q-orbits of D(C. Recall that we are considering as default ordering that given by X 1 < < X n. For every i {1,..., n 1}, let us denote by D i D i (C, the image of the projection onto the first i-coordinates of D D(C, respectively. For any 1 i n 1 e D i, we define R 0 (e = { a Z ri+1 (e, a D i+1 } R(e = { a Z ri+1 (e, a D i+1 (C }. Under this notation, e D i (C if only if R(e, in this case R(e = R(e, where R(e has been defined in (2. Again, the algorithm is based on the computation of n families of sequences of natural numbers. For each e D n 1, set

7 7 M(e = a b R(e m(e, a. Note that, from (3 we have that e D n 1 (C if only if 0 M(e = M(e. Then, we order the different values M( we define So we obtain a sequence as in (4 f [1] = max { M(e} e D n 1 f [i] = max { M(e M(e < f [i 1]}. e D n 1 r n = f [0] f [1] > > f [t] f [t + 1] = 0, (9 where f [0] = r n f [t + 1] = 0 by definition. In this case, t = s if only if D n 1 = D n 1 (C t = s + 1, otherwise, that is, if only if, there exists e D n 1 \ D n 1 (C for which M(e = 0. The sequence given in (9 is the initial one for any value of n it defines the first family. Now, suppose that n 3. Then, for each 0 u t e D n 2 we define Ω 0 (e =, } Ω u (e = {a R 0 (e M(e, a f [u], whenever u 0, µ u (e = a b Ω u(e m(e, a. Note that µ 0 (e = 0 µ t (e = r n 1, for all e D n 2, it may happen that µ u (e = 0 even if e D n 2 (C. Then we define f [u, 1] = max { µ u (e} e D n 2 f [u, i] = max { µ u (e µ u (e < f [u, i 1]}. e D n 2 Then we obtain the family of sequences (one for each u {0,..., t} { r n 1 = f [u, 0] f [u, 1] > > f [u, t(u] f [u, t(u + 1] = 0 0 u t }, where f [u, t(u] denotes the minimum value of the parameters µ u ( the equalities f [u, t(u + 1] = 0, f [u, 0] = r n 1 are given by definition. In a similar way to Section III we describe how to define a family of sequences from the previous ones. Suppose that we have constructed the j-th family, when n 1 > j 1. For the sake of brevity, in what follows we denote δ = n j + 2. { r δ 1 = f [u n,..., u δ, 0] f [u n,..., u δ, 1] >... f [u n,..., u δ, t(u n,..., u δ ] > 0 = f [u n,..., u δ, t(u n,..., u δ + 1] (u n,..., u δ Υ } j (C, where, for every i = 2,..., n, Υ i (C = {(u n,..., u n i+2 0 u n t (10 0 u δ t(u n,..., u δ+1 for δ = n i + 2,..., n 1}. For every (u n,..., u n j+2 Υ j (C we take the corresponding sequence r δ 1 = f [u n,..., u δ, 0] f [u n,..., u δ, 1] >... f [u n,..., u δ, t(u n,..., u δ ] > 0 = f [u n,..., u δ, t(u n,..., u δ + 1].

8 8 Take u {0,..., t(u n,..., u δ }. Then, for every e D n j 1 we define Ω un,...,u δ,0(e =, Ω un,...,u δ,u(e = {a R 0 (e µ un,...,u δ (e, a f [u n,..., u δ, u]}, (11 whenever u 0, µ un,...,u δ,u(e = m(e, a. a b Ω un,...,uδ,u(e By ordering the values above, we get r n j = f [u n,..., u δ, u, 0] f [u n,..., u δ, u, 1] >... f [u n,..., u, t(u n,..., u] f [u n,..., u, t(u n,..., u + 1] = 0, with the first last equalities given by definition. Then we obtain the (j + 1-th family of sequences { r n j = f [u n,..., u n j+1, 0] f [u n,..., u n j+1, 1] >... f [u n,..., u n j+1, t(u n,..., u n j+1 ] f [u n,..., u n j+1, t(u n,..., u n j+1 + 1] = 0, (u n,..., u n j+1 Υ j+1 (C }. We follow the previous process until to get n 1 families of sequences. Then, by using all the previous ones, we define the last family. For every (u n,..., u 2 Υ n (C we set ĝ[u n,..., u 2 ] = e D 1 cm(e ˆf[u 2 ] 0 if u 2 = 0 m(e if n = 2, u 2 0 m(e if n > 2, u 2 0 (12 e D 1 bµun,...,u 3 (e ˆf[un,...,u 2 ] Finally, as in Section III, we define Γ(C = {(i 1,..., i n Z r1 Z rn (13 there exists (u n,..., u 2 Υ n (C such that f [u n,..., u j + 1] i j < f [u n,..., u j ], for j = 2,..., n, 0 i 1 < ĝ[u n,..., u 2 ]}. Example 7. Let q = 2, n = 2, r 1 = 7, r 2 = 5, consider the abelian code C given in Example 6. In this case we choose D = {(0, 0, (0, 1, (1, 0, (1, 1, (3, 0, (3, 1} as restricted representatives of the 2-orbits of D = Z 7 Z 5. Then D(C = {(0, 0, (1, 1}. Then we compute M(0 = 1, M(1 = 4, M(3 = 0 m(0 = 1, m(1 = 3, m(3 = 3. Using these values we get the new sequences f [0] = 5 > f [1] = 4 > f [2] = 1 > f [3] = 0 = f [4] ĝ[0] = 0 < ĝ[1] = 3 < ĝ[2] = 4 < ĝ[3] = 7. Now, following (13, we get the picture

9 9 f [0] = 5 f [1] f [2]..... f [4] = = f... [3] = 0. ĝ[1] ĝ[2] ĝ[3] = 7 ĝ[0] The reader may check that even though we have added four marks, all of them are superfluos, that is, this picture is the same as that of Example 6. Now we shall see that, in fact, the sets Γ(C Γ(C are the same. Let us recall that the values t( s( represent the lengths of the sequences involved in the construction of Γ(C. Lemma 8. Let C be an abelian code in A(r 1,..., r n. Let (v n,..., v 2 Υ n (C (see (5. Then a t = s or s + 1 f [v n] = f [v n] 0, for every j = 2,..., n 1, the following conditions hold: b t(v n,..., v j+1 = { s(v n,..., v j+1 or s(v n,..., v j+1 + 1, c µ vn,...,v j+1 (e = µvn,...,v (e if e D j+1 j 1(C, 0 if e D j 1 \ D j 1 (C, d f [v n,..., v j ] = f [v n,..., v j ] 0. Moreover, e g[v n,..., v 2 ] = ĝ[v n,..., v 2 ] 0. Proof: Let D(C denote the defining set of C. Take D a restricted set of representatives of the q-orbits of Z r1 Z rn. Then we choose D(C = D D(C as a set of restricted representatives of the q-orbits of D(C. First, note that given e D n 1, we have that e D n 1 (C if only if M(e 0, in this case M(e = M(e. This implies that t = s if only if D n 1 = D n 1 (C t = s + 1 otherwise. Moreover, in the former case, f [t] = f [t + 1] = 0. So, for all u {1,..., s} one has that 0 f [u] = f [u] f [u + 1] = f [u + 1]. This gives us a. Now, take (v n,..., v 2 Υ n (C. We are going to prove that b, c d hold for every j = 2,..., n 1. We use induction on n j. First, we prove the case j = n 1. As we have seen, f [vn] = f [v n] 0, so, for every e D n 2, Ω vn (e = in case e D n 2 \ D n 2 (C Ω vn (e = Ω vn (e in case e D n 2 (C. That is, µ vn (e = 0 if e D n 2 \ D n 2 (C µ vn (e = µ vn (e if e D n 2 (C. Then, for all u {1,..., s(v n } one has that f [v n, u] = f [v n, u] 0; in particular, f [v n, v n 1 ] = f [v n, v n 1 ] 0. Moreover, t(v n = s(v n if only if Ω vn (e for every e D n 2, t(v n = s(v n + 1 otherwise. This shows the case j = n 1. Assume that we have proved b, c d for some j {3,..., n 1}. Then, let us prove these conditions in the case j 1. Applying c d with j, we have that Ω vn,...,v j (e = Ω vn,...,v j (e for every e D j 2 (C, Ω vn,...,v j (e = if e D j 2 \ D j 2 (C. Hence, we obtain that µ vn,...,v j (e = µ vn,...,v j (e, for every e D j 2 (C µ vn,...,v j (e = 0 if e D j 2 \ D j 2 (C. Then f [v n,..., v j, u] = f [v n,..., v j, u] 0, for any u {1,..., s(v n,..., v j }; in particular, f [v n,..., v j 1 ] = f [v n,..., v j 1 ] 0. This shows b, c d. In addition, we have obtained that the equality t(v n,..., v j = s(v n,..., v j holds if only if Ω vn,...,v j (e for every e D j 2 t(v n,..., v j = s(v n,..., v j + 1 otherwise. Finally d follows from the definitions (12 (6 from c d with j = 2 (if n = 2, we use that M(e = M(e 0 if only if e D n 1 (C. Proposition 9. Let C be an abelian code in A(r 1,..., r n. Let Γ(C Γ(C the sets of check positions defined in (8 (13 respectively. Then Γ(C = Γ(C. Proof: Let D(C be the defining set of C. Recall that the set Γ(C does not depend on the choice of the restricted set of representatives of the q-orbits of D(C. Take D a restricted set of representatives of the q-orbits of Z r1 Z rn. Then we choose D(C = D D(C as a set of restricted representatives of the q-orbits of D(C we construct Γ(C Γ(C. First, let us note that by using a b in Lemma 8 we have that Υ n (C Υ n (C. Indeed, if (v n,..., v 2 Υ n (C then 0 v n s t 0 v i s(v n,..., v i+1 t(v n,..., v i+1, with i = 2,..., n 1. This implies that (v n,..., v 2 Υ n (C.

10 10 Now, we are going to prove that Γ(C Γ(C. To do this we take (i 1,..., i n Γ(C. Then there exists (v n,..., v 2 Υ n (C such that f [v n,..., v j ] > i j f [v n,..., v j + 1], for every j = 2,..., n, 0 i 1 < g[v n,..., v 2 ]. So, by the previous paragraph, (v n,..., v 2 Υ n (C. By applying a, d e in Lemma 8 we have that f [v n,..., v j ] > i j f [v n,..., v j + 1], for every j = 2,..., n, 0 i 1 < ĝ[v n,..., v 2 ]. Therefore, (i 1,..., i n Γ(C. Let us see the inclusion Γ(C Γ(C. Let (i 1,..., i n Γ(C. Then there exists (v n,..., v 2 Υ n (C such that for every j = 2,..., n, f [v n,..., v j ] > i j f [v n,..., v j + 1], 0 i 1 < ĝ[v n,..., v 2 ]. We will prove that (v n,..., v 2 Υ n (C. We claim that v j 0 for every j = 2,..., n. Assume that this is not true let j 0 be the minimum in {2,..., n} such that v j0 = 0. Since 0 i 1 < ĝ[v n,..., v 2 ] ĝ[v n,..., v 3, 0] = 0, by the definition (12, we have that j 0 > 2. Then, by using (11 we have that µ vn,...,v j0 +1,0(e = 0, for every e D j0 2, hence t(v n,..., v j0+1, 0 = 1. So we get the following sequence r j0 1 = f [v n,..., v j0 +1, 0, 0] > f [v n,..., v j0 +1, 0, 1] = = f [v n,..., v j0 +1, 0, 2] = 0. This implies that f [v n,..., v j0 +1, 0, 1] i j0 1 < f [v n,..., v j0 +1, 0, 0] an so v j0 1 = 0. This contradicts the minimality of j 0. Then our claim is proved. Finally we are going to show that v n s + 1 v j s(v n,..., v j+1 + 1, for every j = 2,..., n. Suppose that there exists j 0 the minimum in {2,..., n} such that eihter v n = s + 1, in case j 0 = n, or v j0 = s(v n,..., v j If j 0 = n then v n = s + 1 so, by using Lemma 8 a, we have that v n = t 0 = f [t + 1] = f [t]. This is a contradiction because f [t + 1] i n < f [t] = f [s + 1] = 0. Let us assume that j 0 < n, then v j0 = s(v n,..., v j = t(v n,..., v j0+1. Hence 0 = f [v n,..., v j0 +1, t(v n,..., v j0 +1] = f [v n,..., v j0 +1, t(v n,..., v j ]. This contradicts that f [v n,..., t(v n,..., v j ] i j0 < f [v n,..., t(v n,..., v j0 +1] = f [v n,..., s(v n,..., v j ] = 0. Thus, we have proved that (v n,..., v 2 Υ n (C. Now, by applying a, d e in Lemma 8 we have that f [v n,..., v j ] > i j f [v n,..., v j + 1], for every j = 2,..., n, 0 i 1 < g[v n,..., v 2 ]. This implies that (i 1,..., i n Γ(C we are done. Let C be an abelian code in A(r 1,..., r n with defining set D(C with respect to certain choice of roots of unity. Then we define D(C = {(r 1 e 1,..., r n e n (e 1,..., e n D(C}, we denote by C 1 the code with defining set D(C with respect to the same set of roots of unity, that is, D(C 1 = D(C. One may check that D(C D(C have the same q-orbits structure, so they yield the same parameters m( (see (1. Therefore, Γ(C = Γ(C 1 (see (8. The following lemma is well known. It establishes the relationship between the defining sets of C 1 C respectively. The reader may find a proof in [13, p. 836]. Lemma 10. Let α = (α 1,..., α n be a choice of primitive roots of unity. Let C be an abelian code in A(r 1,..., r n with defining set D eα (C, let C denote the dual code. Then D eα (C = (Z r1 Z rn \ D eα (C 1. Given C an abelian code in A(r 1,..., r n with defining set D(C with respect to certain roots of unity, we denote by C the abelian code with defining set D(C = (Z r1 Z rn \ D(C with respect to the same choice of roots of unity. The following result follows from Lemma 10 what we have mentioned above about C 1. Corollary 11. Let C be an abelian code in A(r 1,..., r n let C be its dual code. For a fixed set of primitive roots of unity we have that Γ(C = Γ(C. The two previous results say that in order to study the relationship between Γ(C Γ(C we can use the sets Γ(C Γ(C. The advantage of using the code C instead of C relies on the fact that we can compute the defining set of C C at the same time. This allows us to easily relate the respective sets of check positions.

11 11 Let us denote by f [ ], ĝ [ ], M (, Ω (, µ (, t t (, the parameters used in the construction of Γ(C. The Lemma 13 establishes the relationship between these new parameters that used in the construction of Γ(C. First, we need to introduce the following recursive notation. Notation 12. Given (u n,..., u 2 Υ n (C (see (10 we define ω 1 (u n = t u n. Suppose that we have defined ω i 1 (u n,..., u n i+2, with 2 i < n. Then we write (δ = n i + 2 We also set ω i (u n,..., u δ 1 = [ω i 1 (u n,..., u δ, t (ω i 1 (u n,..., u δ u δ 1 ]. ω + i (u n,..., u δ 1 = ω i (u n,..., u i, u δ 1 1 = [ω i 1 (u n,..., u δ, t (ω i 1 (u n,..., u δ u δ 1 + 1]. Lemma 13. Let C be an abelian code in A(r 1,..., r n. Then, for every (u n,..., u 2 Υ n (C, we have that a t = t y t(u n,..., u i = t (ω n i+1 (u n,..., u i, with i = 3,..., n, b f [un,..., u i ] = r i f [ω + n i+1 (un,..., u i], with i = 2,..., n, c ĝ[u n,..., u 2 ] = r 1 ĝ [ω n 1 (u n,..., u 2 ]. Proof: Let α = (α 1,..., α n be a choice of primitive roots of unity. Let us denote D(C = D eα (C D(C = D eα (C, the defining sets of C C respectively, with respect to α. Let D be a set of restricted representatives of the q-orbits of Z r1 Z rn. Take D(C = D D(C D(C = D D(C as sets of restricted representatives of the q-orbits of D(C D(C respectively. Then one has that D(C D(C = D, where the union is disjoint. Let us recall that for every i {1,..., n 1}, the images of the projections onto the first i coordinates of D, D(C D(C are denoted by D i, D i (C D i (C, respectively. Then for every e D n 1, we have that M(e + M (e = r n. This implies that t = t so, by the definitions of f [ ] f [ ], we obtain b in case i = n. In order to prove the remaining cases in a b, we use induction on n i. The key is to prove that for every i {3,..., n}, (u n,..., u i Υ n i+2 (C e D i 2 the following condition holds µ un,...,u i (e + µ ω n i+1(u n,...,u i(e = r i 1. (14 Take (u n,..., u 2 Υ n (C. We are going to prove (14 in case i = n consequently we will obtain b with i = n 1 a with i = n. Let e D n 2. First, suppose that u n 0. For every a R 0 (e, we have that a R 0 (e \ ( Ωun (e if only if M(e, a < f [u n]. Since we have proved b in case i = n, we have that a R 0 (e \ ( Ωun (e if only if M (e, a > f [ω + (un]. This last inequality is equivalent to a Ω 1 ω 1(u n(e because f [ω 1 (u n] > f [ω + 1 (un] = f [ω 1 (u n + 1]. So, R 0 (e \ ( Ωun (e = Ω ω 1(u n(e. Now, suppose that u n = 0. Then Ω 0 (e = ω 1 (0 = t = t. By the definition of t, we have that M (e, a f [t ] for every a R 0 (e. So R 0 (e \ ( Ω0 (e = R 0 (e = Ω ω 1(0(e. This implies that for every u n {0,..., t} e D n 2 the condition (14 with i = n holds. Hence t(u n = t (ω 1 (u n f [u n, u n 1 ] + f [ω + 2 (un, u n 1] = r n 1. This gives us a with i = n b with i = n 1. Suppose that we have proved (14 for i = i 0 + 1, where 2 i 0 < n. Then f [u n,..., u i0 ] + f [ω + n i 0 +1 (un,..., u i 0 ] = r i0 (15 t(u n,..., u i0+1 = t (ω n i0 (u n,..., u i0+1, (16 that is, we have b with i = i 0 a for i = i Let us prove (14 with i = i 0 we will obtain b with i = i 0 1 a with i = i 0.

12 12 Take e D i0 2. We distinguish in two cases again. First, suppose that u i0 0. Then a R 0 (e \ ( Ωun,...,u (e i0 if only if µ un,...,u (e, a < f i0 [u +1 n,..., u i0 ], By using (14, with i = i 0 + 1, (15, this is equivalent to µ ω n i0 (u n,...,u i0 +1(e, a > f [ω + n i 0 +1 (un,..., u i 0 ], this occurs if only if a Ω ω n i0 +1(u n,...,u i0 (e, because f [ω n i0 +1(u n,..., u i0 ] > f [ω + n i 0 +1 (un,..., u i 0 ]. Now, suppose that u i0 = 0. Then, Ω un,...,u i0 +1,0(e = ω n i0+1(u n,..., u i0+1, 0 = [ω n i0 (u n,..., u i0+1, t (ω n i0 (u n,..., u i0+1]. By the definition of t (ω n i0 (u n,..., u i0+1, for every a R 0 (e we have that so µ ω n i0 (u n,...,u i0 +1(e, a f [ω n i0 (u n,..., u i0 +1, t (ω n i0 (u n,..., u i0 +1]; R 0 (e \ ( Ωun,...,u i0,0 (e = R 0 (e = Therefore, we conclude that for every e D i0 2 we have that Ω ω n i0 +1(u n,...,u i0 +1,0(e. µ un,...,u i0 (e + µ ω n i0 +1(u n,...,u i0 (e = r i0 1 then t(u n,..., u i0 = t (ω n i0+1(u l,..., u i0 f [u n,..., u i0 1] + f [ω + n i 0 +2 (un,..., u i 0 1] = r i0 1. This proves (14, a b. To finish the proof let us deal with c. From the previous paragraphs, µ un,...,u 3 (e + µ ω n 2(u n,...,u 3(e = r 2 f [u n,..., u 2 ] + f [ω + n 1 (un,..., u 2] = r 2, then So we are done. r 1 = = = m(e = e D 1 m(e + {e D 1 bµun,...,u 3 (e bf[un,...,u 2 ]} {e D 1 bµun,...,u 3 (e bf[un,...,u 2 ]} m(e + {e D 1 bµun,...,u 3 (e< bf[un,...,u 2 ]} m(e m(e {e D 1 bµ ωn 1 (un,...,u 3 (e c f [ω n 1 (un,...,u 2 ]} = ĝ[u n,..., u 2 ] + ĝ [ω n 1 (u n,..., u 2 ]. The following theorem is the main result of this section. It gives us the relationship between Γ(C Γ(C. Theorem 14. Let C be an abelian code in A(r 1,..., r n. Let κ : n i=1 Z r i n i=1 Z r i be the bijection given by κ(i 1,..., i n = (r 1 i 1 1,..., r n i n 1. Then κ (Γ(C = (Z r1 Z rn \ Γ(C.

13 13 Proof: Let (i 1,..., i n Γ(C = Γ(C. Then, there is a list (u n,..., u 2 Υ n (C satisfying that for j = 2,..., n, For every j = 2,..., n we write κ j (i j = r j i j 1. Then f [u n,..., u j + 1] = f [u n,..., u j + 1] i j < f [u n,..., u j ] = f [u n,..., u j ], 0 i 1 < ĝ[u n,..., u 2 ] = g[u n,..., u 2 ]. r j f [u n,..., u j ] 1 < κ j (i j r j f [u n,..., u j + 1] 1 r 1 ĝ[u n,..., u 2 ] 1 < κ 1 (i 1 r 1 1. Then, by applying Lemma 13 we have that for each 2 j n, f [ω + n j+1 (un,..., u j] κ j (i j < f [ω + n j+1 (un,..., u j + 1], (17 ĝ [ω n 1 (u n,..., u 2 ] κ 1 (i 1 < r 1. (18 Now, note that ω + n j+1 (u n,..., u j + 1 = ω n j+1 (u n,..., u j. So, by (17 (13, (κ 1 (i 1,..., κ n (i n Γ(C if only if κ 1 (i 1 < ĝ [ω n 1 (u n,..., u 2 ]. But this contradicts (18. Hence κ (Γ(C Γ(C c. Finally, from the fact that Γ(C = D(C (see [3] Corollary 11 we have that Γ(C c = Γ(C c = = n r i D(C = i=1 ( n n r i r i D(C = Γ(C, i=1 i=1 which yields the reverse inclusion. Note that the bijection κ may be extended to a bijection of A(r 1,..., r n via κ(p (X 1,..., X n = a κ 1 (jx j, for every P (X 1,..., X n = a j X j. We also denote by κ this extension. It is clear that κ is exactly the composition T r1 1 1 Tn rn 1 (see Section II for notation so it belongs to the permutation automorphism group of every abelian code. Then, Theorem 14 implies that we can use equivalently Γ(C c or Γ(C in order to apply the algorithm of permutation decoding. Indeed, the reader may check that P T i i = 1,..., n is a partial PD-set for C the information set Γ(C c if only if κ P κ 1 is a partial PD-set for C the information set Γ(C. V. PARTIAL PD-SETS FOR ABELIAN CODES In the following two sections we study how to apply the permutation decoding algorithm to abelian codes by taking as reference the information sets given in the previous sections. Since these information sets depends only on their defining sets, we may construct good abelian codes (in order to apply the permutation decoding algorithm from a suitable choice of their defining sets (see Section VI. We shall show sufficient conditions to find (partial PD-sets contained in the subgroup of the permutation automorphisms of any abelian code, generated by the translations T j (i 1,..., i n = (i 1,..., i j + 1,..., i n, for j = 1,..., n, which were introduced in Section II. As we have seen, for a given abelian code C in A(r 1,..., r n both sets Γ (C c Γ ( C (= Γ (C are information sets that we may use equivalently. We prefer to make use of the set Γ (C because, on the one h, we think that in this context the proofs may be written in a simpler form on the other h, examples may be also constructed in an easier way, as the reader may check in next section. As we have noted in the previous section, we can compute the defining sets of C C simultaneously; we can take one or the other as it suits us. We denote by f [ ], g [ ], M (, Ω (, µ (, s s (, the parameters used in the construction of Γ(C. It may be necessary to revise this construction in Section III, more precisely the definitions from (4 to (8. Let C be an abelian t-error-correcting code in A(r 1,..., r n with defining set D(C information set Γ(C. Let b t a natural number let e A(r 1,..., r n an error vector with supp(e = {p 1,..., p b } Z r1 Z rn. We set p j = (p 1 j,..., pn j for j = 1,..., b. We are going to use the subgroup {T j } n j=1 to move supp(e outside of Γ(C. Following the notation in Section III, if p j Γ(C, for some 1 j b, then there exists (u n,..., u 2 Υ n (C satisfying f [u n,..., u i + 1] p i j < f [u n,..., u i ], for every i = 2,..., n, g [u n,..., u 2 ] > p 1 j 0. Therefore, in order to move p j outside of Γ(C, we look for a suitable T {T j } n j=1 such that T (pj / Γ(C ; that is, such that for T (p j there no exist (u n,..., u 2 satisfying the conditions mentioned above. (For instance, the existence of i {1,..., n} α N such that f [u n,..., u i ] p i j + α mod r i implies that T α i (p j / Γ(C.

14 14 Remark 15. As we have seen in Section III, our information sets depend on the chosen ordering in the indeterminates X 1,..., X n. In this section we will use the default ordering, that is, X 1 < < X n. The reader may check that all the results can be adapted to any other ordering. First, we need some technical results some additional notation. All throughout C will be an abelian code in A (r 1,..., r n with defining set D(C information set Γ(C. Notation 16. Let v = (v n,..., v j Υ n j+2 (C, with j {2,..., n}. We define S 0 (v = v, S (v = (v, s (v = (v n,..., v j, s (v n,..., v j (if n 3, if we assume that S i (v is defined, where 0 i < j 2, we set S i+1 (v = ( S i (v, s ( S i (v. Finally we define S 1 (v = (v n,..., v j+1, in case j < n, S 1 (v n =. Lemma 17. Let h, j N such that 2 h j n. Then for every (u n,..., u 2 Υ n (C an integer w, such that u j w s (u n,..., u j+1, we have that: a f [u n,..., u h+1, u h ] f [S j 1 h (u n,..., u j+1, w, 1], b g [u n,..., u 2 ] g [S j 2 (u n,..., u j+1, w]. Proof: Let D(C be a restricted set of representatives of D(C. In order to avoid the distinction in cases, we will assume that, for every e D n 1 (C, µ u n,...,u i+1 (e = M (e, when i = n µ (e = M (e. Let (u n,..., u 2 Υ n (C. Let i j be natural numbers such that 2 i j n w {u j,..., s (u n,..., u j+1 }. We are going to prove the following claim: if e is an element of D i 1 (C such that µ u n,...,u i+1 (e f [u n,..., u i ] then µ S j 1 i (u n,...,u j+1,w (e f [S j i (u n,..., u j+1, w]. The condition (19 will be used repeatedly through the proof. First, in case i = j we have that µ S j 1 i (u n,...,u j+1,w (e = µ u n,...,u j+1 (e f [u n;..., u j+1, u, j] f [u n,..., u j+1, w] = f [S j i (u n,..., u j+1, w] becasuse w u j. Assume that i < j (note that this implies i < n that there exists e D i 1 (C such that µ u n,...,u i+1 (e f [u n,..., u i ]. Then µ u n,...,u i+1 (e 0, so there exists a 1 R(e verifying that µ u n,...,u i+2 (e, a 1 f [u n,..., u i+1 ]. Hence µ u n,...,u i+2 (e, a 1 0. By repeating this argument j i times we get j i natural numbers a 1 R(e,..., a j i R(e, a 1,..., a j i 1 such that µ u n,...,u j+1 (e, a 1,..., a j i f [u n,..., u j ]. So, µ u n,...,u j+1 (e, a 1,..., a j i f [u n,..., u j+1, w] because w u j. Then one follows that hence µ u n,...,u j+1,w(e, a 1,..., a j i 1 0, µ u n,...,u j+1,w(e, a 1,..., a j i 1 f [S(u n,..., u j+1, w]. Therefore, µ S(u n,...,u j+1,w (e, a 1,..., a j i 2 0. Now, we repeat this process j i times we remove the numbers a i, with i = 1,..., j i, until to get µ S j i 1 (u n,...,u j+1,w (e f [S j i (u n,..., u j+1, w]. This proves (19. Let us prove a. Let h be a natural number with 2 h j n. If h = j, then the proof of a is straightforward. So suppose that h < j. Then one has f [u n,..., u h+1, u h ] f [u n,..., u h+1, 1] = { } max µ u n,...,u h+1 (e = e D h 1 (C max m(e, a e D h 1 (C. a Ω un,...,u h+1 (e (19

15 15 Recall that On the other h, Ω u n,...,u h+1 (e = {a R(e µ u n,...,u h+2 (e, a f [u n,..., u h+1 ]}. f [S j 1 h (u n,..., u j+1, w, 1] = } max {µ S j 1 h(un,...,uj+1,w (e = e D h 1 (C max m(e, a e D h 1 (C, a Ω S j 1 h (un,...,u j+1,w (e where Now, by applying (19 with i = h + 1 we have a. Finally let us prove b. By definition, Ω S j 1 h (u n,...,u j+1,w (e = {a R(e µ S j 2 h (u n,...,u j+1,w (e, a f [S j 1 h (u n,..., u j+1, w]}. g [u n,..., u 2 ] = Then by taking v = (u n,..., u j+1, w we have that g [S j 2 (v] = µ un,...,u 3 (e f [u n,...,u 2] µ S j 3 (v (e f [S j 2 (v] Therefore, by (19 with i = 2 we obtain b. Now we present some definitions that will be used in our main results. Definition 18. Given v {1,..., s } we define m(e. m(e. Λ(v = {i {2,..., n} f [S n i 1 (v, 1] < r i }, where, if i = n then f [S 1 (v, 1] = f [1]. From Λ(v we define { Λ(v if g [S Λ(v = n 2 (v] = r 1, Λ(v {1} otherwise. Now, for every i Λ(v, we define in case i > 1, in case i = 1 we set { λ i (v = max λ N f [S n i 1 (v, 1] < { λ 1 (v = max λ N g [S n 2 (v] < Note that, by Lemma 17, given v, v {1,..., s } we have that ri }, λ r1 }. λ if v v then Λ(v Λ(v. (20 The following two lemmas will be used several times in the proofs of the subsequent results. Given r > 0 x integers, we denote by [x] r the remainder of the division of x between r. Lemma 19. Let r, h, x 1,..., x h be natural numbers such that 0 x 1 < x 2 < < x h < r. Then there exists β N verifying that r h 1 [x i + β] r < r, for every i = 1,..., h, [x j + β] r = r 1 for some j {1,..., h}.

16 16 Proof: For every i {2,..., h} we denote d i = x i x i 1 we set d 1 = r (x h x 1. So r = h i=1 d i. Suppose that d i < r h for every i = 1,..., h. Then di < r h, for every i = 1,..., h. Hence h r = d i < h r h = r, i=1 a contradiction. Therefore there exists k {1,..., h} such that d k r h. Take β = r xk 1 1, where we assume x 0 = x h. Then, on the one h, we have that x i + β < 2r, for every i = 1,..., h. On the other h, if k 1 then r x k + β β x k x k 1 1 1, h if k = 1 then So, one has that we are done. x 1 + β = r x h 1 + x 1 = d 1 1 r h 1. 0 r h 1 x1 + β < < x k 1 + β = r 1 < r r + r h 1 xk + β < < x h + β < 2r, Lemma 20. Let B be a subset of Z r1 Z rn with cardinality b. Let A B {λ i (u} i I a set of integers, with I Λ(u for certain u {1,..., s }. For any non negative integer η such that η A η + i I λ i (u b, there exist J = {i 1,..., i h } I a family of sets {X k } h k=1 (X k Z rk such that where the union is disjoint, for each k {1,..., h}, B = A B 1 B h, B k = {x B \ (A B 1 B k 1 π ik (x X k }. Moreover, there exists a family of non negative integers {β k } h k=1 verifying that for every k = 1,..., h, given a X k v = (v n,..., v ik +1 Υ n ik +1(C, where v n u, the following properties hold a [a + β k ] rik f [v, 1], si i k 1, b [a + β 1 ] r1 g [v], si i k = 1. Proof: Suppose that we have A B, {λ i (u} i I η satisfying the conditions of the statement. First of all, note that if A = B then the lemma follows strightforward by taking J =. So, we assume that A B. We denote λ i (u = λ i, for short, the elements of I by I = {i 1,..., i m }. We are going to construct J = {i 1,..., i h }, where we will get h the family of sets {X k } h k=1 as a consequence of a recursive procedure. To do this we will make use of an intermediate family of sets {L k } h k=1, with L k Z rk. For each k {1,..., m} we define L k = π ik [B \ (A B 1 B k 1 ] ; { X Lk such that X = λ X k = ik, if L k > λ ik, L k, if L k λ ik, B k = {x B \ (A B 1 B k 1 π ik (x X k }. We set h = min{1 k m L k λ ik }. Let us see that {1 k m L k λ ik } =, that is, let us check that h is well defined. Suppose that this is not true. Then we have that L m > λ im. This implies that B \ (A B 1 B m = B ( A + m k=1 B k > 0, because X m = λ im < L m. So m m η + λ ik A + B k < B = b, a contradiction. Therefore L h λ ih, then X h = L h. Hence { ( } h 1 B h = x B \ A B k π ih (x L h = k=1 k=1 k=1 = B \ ( A h 1 k=1 B k,

17 17 so B = A B 1 B h. Now, for every k {1,..., h}, we use Lemma 19 applied to the elements of each X k r ik. Then we get β 1,..., β h N such that rik rik 1 1 [a + β k ] rik < r ik X k λ ik for all a X k. Note that X k λ k for every k = 1,..., h. On the other h, by using that i k Λ(u for every k = 1,..., h, Lemma 17, we have that for every a X k v = (v n,..., v ik +1 Υ n ik +1(C, with v n u, [a + β k ] rik f [S n i k 1 (u, 1] f [v, 1], (21 in case i k 1, [a + β k ] r1 g [S n 2 (u] g [v], (22 in case i k = 1. This finishes the proof. Now, we are ready to present our first sufficient condition to have a (partial PD-set contained in the group T i i = 1,..., n. Given p in Z r1 Z rn, for every i = 1,..., n, we denote by π i (p the i-th coordinate of p. In general, if B is a subset of Z r1 Z rn, then π i (B denotes the set of i-th coordinates of the elements of B. Theorem 21. Let C be a t-error-correcting abelian code in A(r 1,..., r n. Let b a positive integer such that b t. If there exists a subset I Λ(s (see (18 verifying λ i (s b then the group T i i I is a b-pd-set for C with respect to the information set Γ(C. i I Proof: Let B a subset in Z r1 Z rn with cardinality b. Suppose that there exists I = {i 1,..., i m } contained in Λ(s such that m k=1 λ i k (s b. We denote λ ik (s = λ ik for brevity. Then by applying the Lemma 20 to B, A = {η, λ i1,..., λ im }, with η = 0, we obtain a family of sets X 1 Z ri1,..., X h Z rih, with h m, satisfying that B = B 1 B h, where, for every k {1,..., h} k 1 B k = x B \ B j π ik (x X k. j=1 Moreover, there exists a family of positive integers {β k } h k=1 such that for every k = 1,..., h, given a X k v = (v n,..., v ik +1 Υ n ik +1(C one has that [a + β k ] rik f [v, 1], (23 in case i k 1, [a + β 1 ] r1 g [v], (24 in case i k = 1. Note that every (v n,..., v ik ( +1 Υ n ik +1(C satisfies that v n s. h Now, take p B let us see that j=1 T βj i j (p / Γ(C. Let k {1,..., h} such that p B k. Let us denote ( h q = j=1 T βj i j (p suppose that q Γ(C. Then there exists v = (v n,..., v 2 Υ n (C such that in case i k 1, in case i k = 1. On the other h, π ik (q = π ik (T β k i k (p f [v n,..., v ik ] > π ik (q f [v n,..., v ik + 1], g [v n,..., v 2 ] > π 1 (q 0, = [π ik (p+β k ] rik by definition π ik (p X k. So, in cases i k 1 i k = 1 we reach a contradiction with (23 (24 respectively (observe that f [v n,..., v ik +1, v ik ] f [v n,..., v ik +1, 1]. The previous theorem is based on the existence of certain subsets of Λ(s. The following result deals with subsets Λ(u, for some u {1,..., s }. So, in some sense it improves Theorem 21 (see (20. Although, the following theorem implies a growth of the (partial PD-set. More precisely, it supposes the addition of the subgroup T n. This means that in some cases we might obtain PD-sets bigger than that given by the Theorem 21. Theorem 22. Let C be a t-error-correcting abelian code in A(r 1,..., r n. Let b be a positive integer such that b t. We define 0 if f [1] = r n, λ 0 = 1 if rnb f [1] < rn, b if f [1] < r nb.