Partial Permutation Decoding for abelian codes
Jos´e Joaqu´ın Bernal and Juan Jacobo Sim´on
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 at-error correcting abelian code to have ab-PD-set for everyb≤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] and it is fully described in [9] and [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 and parity check matrix in standard 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 and only if all its information symbols are correct.
Finding adequate information sets and 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 and 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 and 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 and 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 and 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 and permutation decoding. In Section III we reproduce without proofs the construction of sets of check positions (and 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 codeCand its dualC⊥. Then, as a first part of our main results, we study the relationship between them, which we denote by Γ(C)and Γ(C⊥)respectively. More precisely, we show that there exists a simple bijection from the complementary set Γ(C)c toΓ(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 Matem´aticas, Universidad de Murcia, 30100 Murcia. Spain.
email: josejoaquin.bernal@um.es, jsimon@um.es
Partially supported by MINECO (Ministerio de Econom´ıa y Competitividad), (Fondo Europeo de Desarrollo Regional) project MTM2012-35240 and Fundaci´on S´eneca of Murcia.
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 and Proposition 26). Finally, Section VI contains applications that show us how we may use the conditions obtained to design and exhibit codes that improve the parameters of best known permutation decodable abelian codes for certain lengths.
II. PRELIMINARIES
ThroughoutF denotes the field withq elements where qis a power of a prime p. LetC be a linear code of dimension k, and lengthl over the fieldF, that is, a subspace ofFlwith dimensionk. We call the elements ofC codewords. An information set forC is a set of positions{i1, . . . , ik} ⊆ {1, . . . , l}such that restricting the codewords to these positions we get the whole spaceFk. For every codeword the symbols in the positions corresponding to an information set are called information symbols and the other l−k positions are called check positions [13]. A generator matrix forC is ak×l matrixGwhose rows form a basis for C. We say that G is in standard form if it is of the form [Ik | A], where Ik 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∈Fl|u·v = 0 for allu∈ C}. A parity check matrix for C is a generator matrix forC⊥. If G is a generator matrix in standard form, it is easy to check that H = [−AT |Il−k]is a parity check matrix. In this case we say thatH is also in standard form.
As usual, for any codewordc∈ C we denote its support bysupp(c); that is, the set of its non-zero entries. We consider the parametert=d−1
2
, wheredis the minimum distance ofC, that measures the error-correction capability ofC. Then we say that C is an [l, k] t-error-correcting code.
We see the group of permutations on l symbols,Sl, acting on Fl via σ(c1, . . . , cl) = cσ−1(1), . . . , cσ−1(l)
withσ ∈Sl. Then the permutation automorphism group of C is
PAut(C) ={σ∈Sl|σ(C) =C}.
Two linear codesC andC0 are said to be permutation equivalent if there existsσ∈Sl such thatσ(C) =C0. It is easy to see that any linear code is permutation equivalent to a code which has a generator matrix in standard form.
Now, we recall some basic facts about the family of abelian codes and 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'Cr1× · · · ×Crn, withCri the cyclic group of order ri, induces a canonical isomorphism of F-algebras fromFGto
F[X1, . . . , Xn]/hX1r1−1, . . . , Xnrn−1i.
We denote this quotient algebra byA(r1, . . . , rn). So, we identify the codewords with polynomialsP(X1, . . . , Xn)such that every monomial satisfy that the degree of the indeterminate Xi is in Zri, the set of non negative integers less than ri. We write the elements P ∈ A(r1, . . . , rn) as P = P(X1, . . . , Xn) = P
ajXj, where j = (j1, . . . , jn)∈ Zr1× · · · ×Zrn and Xj=X1j1· · ·Xnjn. We deal with abelian codes in the semisimple case, that is, we always assume thatgcd(ri, q) = 1for every i= 1, . . . , n.
Our construction makes use of the structure of roots of the ideals inA(r1, . . . , rn); so let us recall some basic facts about it. For a fixed primitive ri-th root of unityαi in some extension of F,i= 1, . . . , n, every abelian code CinA(r1, . . . , rn)is totally determined by its root set,
Z(C) = {(αa11, . . . , αann) | P(αa11, . . . , αann) = 0for all P(X1, . . . , Xn)∈ C}.
The definingset of C with respect toα={α1, . . . , αn} is
Dα(C) = {(a1, . . . , an)∈Zr1× · · · ×Zrn | (αa11, . . . , αann)∈ Z(C)}.
Given an abelian code C ⊆ A(r1, . . . , rn) with defining set Dα(C) if one chooses different primitive roots of unity, say β ={β1, . . . , βn}, then the set Dβ(C)detemines a new code, sayC0, 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, and we denote the defining set of C byD(C).
Recall that, for γ∈N, the qγ-cyclotomic coset of an integeramoduloris the set C(qγ,r)(a) =
a·qγ·i | i∈N ⊆Zr. We extend the concept of q-cyclotomic coset of an integer to several components.
Definition 1. Given an element(a1, . . . , an)∈Zr1× · · · ×Zrn, we define itsq-orbit modulo (r1, . . . , rn)as Q(a1, . . . , an) =
a1·qi, . . . , an·qi
| i∈N ⊆ Zr1× · · · ×Zrn.
It is easy to see that for every abelian codeC ⊆A(r1, . . . , rn),D(C)is closed under multiplication byqinZr1× · · · ×Zrn, and then D(C) is necessarily a disjoint union of q-orbits modulo (r1, . . . , rn). Conversely, every union of q-orbits modulo (r1, . . . , rn)defines an abelian code inA(r1, . . . , rn). For the sake of simplicity we only writeq-orbit, and 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] and [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. LetI be an information set forC. For s≤t a s-PD-setfor C and I is a subsetP ⊆PAut(C)such that every set of scoordinate positions is moved out ofI by at least one element of P. In cases=t, we say thatP 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 thant are correct.
We denote the Hamming weight of a vector v∈Fl bywt(v).
Theorem 3 ([9], Theorem 8.1). Let C be an[l, k] t-error-correcting code with parity check matrix H in standard form. Let r=c+ebe a vector, wherec∈ Cand wt(e)≤t. Then the information symbols inrare correct if and only ifwt HrT
≤t.
Once we have found a PD-set P ⊆PAut(C) for the given codeC with respect to the information setI, the algorithm of permutation decoding is as follows: take a parity check matrix H for C in standard form. Suppose that we receive a vector r = c+e, where c ∈ C and e represents the error vector and satisfies that wt(e) ≤ t. Then we calculate the syndromes H(τ(r))T, withτ∈P, until we obtain a vectorH(τ0(r))T with weight less than or equal tot. 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 and then we can construct a codewordc0∈ C. Finally, we decode toτ−1(c0) =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 Zr1 × · · · ×Zrn by Sr1×···×rn and we consider it acting on A(r1, . . . , rn) via τ
P
jajXj
=P
jajXτ(j). From this point of view the permutation automorphism group of an abelian codeCinA(r1, . . . , rn) may be described as
PAut(C) ={τ∈Sr1×···×rn | τ(C) =C}.
Let Tj be the transformation from A(r1, . . . , rn) into itself, given byTj(P) =Xj·P, for j = 1, . . . , n. Then it is clear that Tj induces a permutation inSr1×···×rn, which we also denote byTj, viaTj(i1, . . . , in) = (i1, . . . , ij+ 1, . . . , in). Then h{Tj}nj=1i may be viewed as a subgroup of permutation automorphisms for every abelian code in A(r1, . . . , rn). We shall look for PD-sets contained in the subgroup h{Tj}nj=1i.
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 algebraA(r1, . . . , rn)under the assumptionsgcd(ri, q) = 1, for alli= 1, . . . , n, andn≥2. LetDbe a union ofq-orbits modulo(r1, . . . , rn)(see Definition 1). For eachi= 1, . . . , nletDi denotes the projection of the elements of D onto the firsti-coordinates. Then, givene= (e1, . . . , ej)∈Dj, with1≤j≤n, we define
γ(e) =|Q(e)|
and
m(e) =
C(q0,rj)(ej)
, (1)
whereq0 =q, in casej = 1, andq0=qγ(e1,...,ej−1) otherwise.
As we have noted, in the semisimple case every defining set of an abelian code in A(r1, . . . , rn) is a union of q-orbits modulo (r1, . . . , rn). 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. LetDbe a union ofq-orbits modulo(r1, . . . , rn)and fix an orderingXi1<· · ·< Xin. A setDof representatives of theq-orbits of Dis called arestricted set of represantives, with respect to the fixed ordering, if for everye= (e1, . . . , en) and e0 = (e01, . . . , e0n) in D one has that, for all j = 1, . . . , n, the equality Q(ei1, . . . , eij) = Q(e0i1, . . . , e0ij) implies that (ei1, . . . , eij) = (e0i1, . . . , e0ij).
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: X1 <
· · ·< Xn.
Now we describe our construction. Let C ⊆ A(r1, . . . , rn) be an abelian code with defining set D(C). Let D(C) be a restricted set of representatives of the q-orbits inD(C), with respect to the default ordering on the indeterminates. As before, for each1≤i≤n, we denote byDi(C)andDi(C)the projection onto the firsti-coordinates ofD(C)andD(C)respectively.
Given e∈Di(C), let
R(e) ={a∈Zri+1|(e, a)∈Di+1(C)}, (2)
where(e, a) has the obvious meaning; that is, ife= (e1, . . . , ei)then(e, a) = (e1, . . . , ei, a).
For the algorithm we need to calculatenfamilies of sequences of natural numbers. For eache∈Dn−1(C), we define M(e) = X
a∈R(e)
m(e, a) (3)
and consider the set{M(e)}e∈D
n−1(C). Then we denote the different values of theM(e)’s as follows,
f[1] = max
e∈Dn−1(C)
{M(e)} and
f[i] = max
e∈Dn−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(·) and we set f[s+ 1] = 0 by convention. Note that M(e)>0, for alle∈Dn−1(C), by definition.
For any value ofn, this is the initial family of sequences and 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 everye∈Dn−2(C)
Ωu(e) ={a∈R(e)|M(e, a)≥f[u]} and
µu(e) = X
a∈Ωu(e)
m(e, a).
Observe that the setΩu(e)may eventually be the empty set. In this case, the corresponding valueµu(e)will be zero.
We define
f[u,1] = max
e∈Dn−2(C)
{µu(e)} and
f[u, i] = max
e∈Dn−2(C)
{µu(e)|0< µu(e)< f[u, i−1]}.
We order the previous parameters and we get the sequence
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 µu(·) and 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}.
In order to describe how to define a family of sequences from the previous ones, suppose that we have constructed thej-th family, when n−1> j≥1. For the sake of brevity, in what follows we denoteδ=n−j+ 2.
{f[un, . . . , uδ,1]>· · ·> f[un, . . . , uδ, s(un, . . . , uδ)]>
>0 =f[un, . . . , uδ, s(un, . . . , uδ) + 1]|(un, . . . , uδ)
∈Υj(C)}
where, for everyi= 2, . . . , n,
Υi(C) = {(un, . . . , un−i+2)|1≤un≤sand (5)
1≤uδ ≤s(un, . . . , uδ+1)for δ=n−i+ 2, . . . , n−1}.
For each (un, . . . , un−j+2)∈Υj(C)we take the corresponding sequence:
f[un, . . . , uδ,1]>· · ·> f[un, . . . , uδ, s(un, . . . , uδ)]>
>0 =f[un, . . . , uδ, s(un, . . . , uδ) + 1]. Let u∈ {1, . . . , s(un, . . . , un−j+2)}. Then, for everye∈Dn−j−1(C)we define
Ωun,...,uδ,u(e) ={a∈R(e)|µun,...,uδ(e, a)≥f[un, . . . , uδ, u]} and
µun,...,uδ,u(e) = X
a∈Ωun,...,uδ ,u(e)
m(e, a).
By ordering the different values µun,...,uδ,u(e), withe∈Dn−j−1(C), we obtain
f[un, . . . , uδ, u,1] > · · ·> f[un, . . . , uδ, u, s(un, . . . , uδ, u)]
> 0 =f[un, . . . , uδ, u, s(un, . . . , uδ, u) + 1],
wheref[un, . . . , uδ, u, s(un, . . . , uδ, u) + 1]= 0by convention. Then the (j+ 1)-th family of sequences is {f[un, . . . , uδ−1,1]>· · ·> f[un, . . . , uδ−1, s(un, . . . , uδ−1)]>
>0 =f[un, . . . , uδ−1, s(un, . . . , uδ−1) + 1]|(un, . . . , uδ−1)
∈Υj+1(C)}.
We follow the previous process until we get n−1families of sequences. Finally, by using all the previous ones, we define, for any value of n, the last family of sequences. For every(un, . . . , u2)∈Υn(C)we define
g[un, . . . , u2]=
P
e∈D1 (C) M(e)≥f[u2 ]
m(e) ifn= 2,
P
e∈D1 (C) µun,...,u3(e)≥
f[un,...,u2 ]
m(e) ifn >2. (6)
So the last family of sequences is
{g[un, . . . , u3,1]<· · ·< g[un, . . . , u3, s(un, . . . , u3)]< (7)
< g[un, . . . , u3, s(un, . . . , u3)]|(un, . . . , u3)∈Υn−1(C)}. The algorithm yields the following set
Γ(C) = {(i1, . . . , in)∈Zr1× · · · ×Zrn| (8)
there exists (un, . . . , u2)∈Υn(C)such that f[un, . . . , uj+ 1]≤ij< f[un, . . . , uj],
forj= 2, . . . , n, and0≤i1< g[un, . . . , u2]}.
The following theorem, proved in [4], establishes that Γ(C)is a set of check positions forC.
Theorem 5 ([4]). LetC be an abelian code in A(r1, . . . , rn). Assume thatgcd(ri, q) = 1, for everyi= 1, . . . , n, andn≥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.
Example 6. Letq= 2,n= 2, r1= 7,r2= 5, and consider the abelian codeC 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) and (3) we computeM(0) = 1, M(1) = 4 andm(0) = 1, m(1) = 3. Using these values we may get the sequences
f[1]= 4> f[2]= 1> f[3]= 0 and
g[1]= 3< g[2]= 4
which correspond to (4) and (6) respectively. We use them to produce the marks in the following picture.
• • • • . . . -.
• • • . . . . .
• • • . . . . .
• • • . . . . .
. . . .
6
7 5
f[3]= 0 f[2]
f[1]
g[1] g[2]
So, a set of check positions forC 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 inA(r1, . . . , rn), we denote byC⊥ its dual code. By the previous section, we know thatΓ C⊥ and the complementary set Γ(C)c ⊆Qn
i=1Zri 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∈ Sr1×···×rn such that for any abelian code C, κ ∈ PAut(C) and, 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 and 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 setD(C) with respect to certain primitive roots of unity. To avoid any confusion with the original definition ofΓ(C)we will make a change on the symbols used to determine it. The new set determined by the new parameters will be denoted byΓ(C). Later, we will see that they coincide.b
Now we choose a set of restricted representativesDof theq-orbits of the whole setD=Zr1×. . .×Zrn 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 theq-orbits of D(C). Recall that we are considering as default ordering that given by X1 <· · ·< Xn. For every i∈ {1, . . . , n−1}, let us denote by Di andDi(C), the image of the projection onto the first i-coordinates of D and D(C), respectively. For any 1≤i≤n−1 ande∈Di, we define
R0(e) =
a∈Zri+1 | (e, a)∈Di+1 and
R(e)b =
a∈Zri+1 | (e, a)∈Di+1(C) .
Under this notation, e∈Di(C)if and only if∅ 6=R(e), and in this caseb R(e) =b R(e), whereR(e)has been defined in (2).
Again, the algorithm is based on the computation ofn families of sequences of natural numbers. For eache∈Dn−1, set
Mc(e) = X
a∈R(e)b
m(e, a).
Note that, from (3) we have that e∈Dn−1(C) if and only if 06=Mc(e) =M(e). Then, we order the different values Mc(·) and we define
fb[1] = max
e∈Dn−1
{Mc(e)} and fb[i] = max
e∈Dn−1
{Mc(e)|Mc(e)<fb[i−1]}.
So we obtain a sequence as in (4)
rn =fb[0]≥fb[1]>· · ·>fb[t]≥fb[t+ 1]= 0, (9) wherefb[0]=rn andfb[t+ 1]= 0by definition. In this case, t=sif and only ifDn−1=Dn−1(C)andt=s+ 1, otherwise, that is, if and only if, there existse∈Dn−1\Dn−1(C)for whichMc(e) = 0.
The sequence given in (9) is the initial one for any value of n and it defines the first family. Now, suppose thatn ≥3.
Then, for each0≤u≤t ande∈Dn−2 we define Ωb0(e) = ∅, Ωbu(e) = n
a∈R0(e) | Mc(e, a)≥fb[u]
o ,
whenever u6= 0, and
µbu(e) = X
a∈bΩu(e)
m(e, a).
Note thatµb0(e) = 0andµbt(e) =rn−1, for alle∈Dn−2, and it may happen that bµu(e) = 0 even ife∈Dn−2(C).
Then we define
fb[u,1] = max
e∈Dn−2
{µbu(e)} and fb[u, i] = max
e∈Dn−2
{µbu(e)|µbu(e)< f[u, i−1]}.
Then we obtain the family of sequences (one for each u∈ {0, . . . , t})
{ rn−1=fb[u,0]≥fb[u,1]>· · ·>fb[u, t(u)]≥ fb[u, t(u) + 1]= 0|0≤u≤t },
where fb[u, t(u)] denotes the minimum value of the parametersµbu(·) and the equalities fb[u, t(u) + 1] = 0,fb[u,0] =rn−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, whenn−1> j≥1. For the sake of brevity, in what follows we denoteδ=n−j+ 2.
n
rδ−1=fb[un, . . . , uδ,0]≥fb[un, . . . , uδ,1]> . . .
fb[un, . . . , uδ, t(un, . . . , uδ)]>0 =fb[un, . . . , uδ, t(un, . . . , uδ) + 1]
|(un, . . . , uδ)∈Υbj(C)o ,
where, for everyi= 2, . . . , n,
Υbi(C) = {(un, . . . , un−i+2)|0≤un≤t and (10)
0≤uδ ≤t(un, . . . , uδ+1)for δ=n−i+ 2, . . . , n−1}.
For every (un, . . . , un−j+2)∈Υbj(C)we take the corresponding sequence rδ−1=fb[un, . . . , uδ,0]≥fb[un, . . . , uδ,1]> . . .
fb[un, . . . , uδ, t(un, . . . , uδ)]>0 =fb[un, . . . , uδ, t(un, . . . , uδ) + 1].
Takeu∈ {0, . . . , t(un, . . . , uδ)}. Then, for everye∈Dn−j−1 we define Ωbun,...,uδ,0(e) = ∅,
Ωbun,...,uδ,u(e) = {a∈R0(e)|µbun,...,uδ(e, a)
≥fb[un, . . . , uδ, u]},
(11)
whenever u6= 0, and
µbun,...,uδ,u(e) = X
a∈bΩun,...,uδ ,u(e)
m(e, a).
By ordering the values above, we get
rn−j=fb[un, . . . , uδ, u,0]≥fb[un, . . . , uδ, u,1]> . . . fb[un, . . . , u, t(un, . . . , u)]≥fb[un, . . . , u, t(un, . . . , u) + 1]= 0,
with the first and last equalities given by definition. Then we obtain the(j+ 1)-th family of sequences { rn−j=fb[un, . . . , un−j+1,0]≥fb[un, . . . , un−j+1,1]> . . .
fb[un, . . . , un−j+1, t(un, . . . , un−j+1)]≥ fb[un, . . . , un−j+1, t(un, . . . , un−j+1) + 1]= 0,
|(un, . . . , un−j+1)∈Υbj+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(un, . . . , u2)∈Υbn(C)we set
bg[un, . . . , u2]=
0 ifu2= 0 X
e∈D1 M(e)≥c f[uˆ 2 ]
m(e) ifn= 2, andu26= 0
X
e∈D1 µun,...,u3b (e)≥
f[un,...,u2 ]ˆ
m(e) ifn >2, andu26= 0
(12)
Finally, as in Section III, we define
Γ(C)b = {(i1, . . . , in)∈Zr1× · · · ×Zrn| (13)
there exists (un, . . . , u2)∈Υbn(C)such that fb[un, . . . , uj+ 1]≤ij<fb[un, . . . , uj],
forj= 2, . . . , n, and0≤i1<bg[un, . . . , u2]}.
Example 7. Let q = 2, n = 2, r1 = 7, r2 = 5, and 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 = Z7×Z5. Then D(C) ={(0,0),(1,1)}. Then we compute Mc(0) = 1, Mc(1) = 4, Mc(3) = 0 andm(0) = 1, m(1) = 3, m(3) = 3. Using these values we get the new sequences
fb[0]= 5>fb[1]= 4>fb[2]= 1>fb[3]= 0 =fb[4]
and
bg[0]= 0<bg[1]= 3<bg[2]= 4<bg[3]= 7.
Now, following (13), we get the picture
6
• • • • . . . -.
• • • . . . . .
• • • . . . . .
• • • . . . . .
. . . .
bg[3]= 7 fb[0]= 5
fb[4]=
=fb[3]= 0 fb[2]
fb[1]
q bg[0]
bg[1] bg[2]
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 setsΓ(C)andbΓ(C)are the same. Let us recall that the valuest(·)ands(·)represent the lengths of the sequences involved in the construction ofΓ(C).
Lemma 8. LetC be an abelian code inA(r1, . . . , rn). Let(vn, . . . , v2)∈Υn(C)(see (5)). Then a) t=sors+ 1and f[vn]=fb[vn]6= 0,
and for every j= 2, . . . , n−1, the following conditions hold:
b) t(vn, . . . , vj+1) =s(vn, . . . , vj+1)ors(vn, . . . , vj+1) + 1, c) µbvn,...,vj+1(e) =
µvn,...,vj+1(e) ife∈Dj−1(C), 0 ife∈Dj−1\Dj−1(C), d) f[vn, . . . , vj]=fb[vn, . . . , vj]6= 0.
Moreover,
e) g[vn, . . . , v2]=bg[vn, . . . , v2]6= 0.
Proof:LetD(C)denote the defining set of C. TakeDa restricted set of representatives of theq-orbits ofZr1× · · · ×Zrn. Then we choose D(C) =D∩D(C)as a set of restricted representatives of theq-orbits ofD(C).
First, note that given e∈Dn−1, we have thate∈Dn−1(C)if and only ifMc(e)6= 0, and in this caseMc(e) =M(e). This implies thatt=sif and only ifDn−1=Dn−1(C)andt=s+ 1otherwise. Moreover, in the former case,fb[t]=fb[t+ 1]= 0.
So, for all u∈ {1, . . . , s} one has that06=f[u]=fb[u]andf[u+ 1]=fb[u+ 1]. This gives us a).
Now, take(vn, . . . , v2)∈Υn(C). We are going to prove thatb),c)andd)hold for everyj= 2, . . . , n−1. We use induction on n−j. First, we prove the case j=n−1. As we have seen,fb[vn]=f[vn]6= 0, and so, for everye∈Dn−2,Ωbvn(e) =∅ in case e ∈ Dn−2\Dn−2(C) and Ωbvn(e) = Ωvn(e) in case e ∈ Dn−2(C). That is, µbvn(e) = 0 if e ∈ Dn−2\Dn−2(C) and µbvn(e) = µvn(e) if e ∈ Dn−2(C). Then, for all u∈ {1, . . . , s(vn)} one has that f[vn, u] =fb[vn, u] 6= 0; in particular, f[vn, vn−1]=fb[vn, vn−1]6= 0. Moreover, t(vn) =s(vn)if and only ifΩbvn(e)6=∅for everye∈Dn−2, andt(vn) =s(vn) + 1 otherwise. This shows the case j=n−1.
Assume that we have proved b), c) and d) for some j ∈ {3, . . . , n−1}. Then, let us prove these conditions in the case j−1. Applyingc)and d) withj, we have that Ωbvn,...,vj(e) = Ωvn,...,vj(e) for everye∈Dj−2(C), and Ωbvn,...,vj(e) =∅ if e∈Dj−2\Dj−2(C). Hence, we obtain that µbvn,...,vj(e) =µvn,...,vj(e), for every e∈Dj−2(C) andbµvn,...,vj(e) = 0if e∈ Dj−2\Dj−2(C). Thenf[vn, . . . , vj, u]=fb[vn, . . . , vj, u]6= 0, for anyu∈ {1, . . . , s(vn, . . . , vj)}; in particular,f[vn, . . . , vj−1]= fb[vn, . . . , vj−1]6= 0. This shows b), c) andd). In addition, we have obtained that the equality t(vn, . . . , vj) =s(vn, . . . , vj) holds if and only ifΩbvn,...,vj(e)6=∅ for everye∈Dj−2 andt(vn, . . . , vj) =s(vn, . . . , vj) + 1otherwise.
Finallyd)follows from the definitions (12) and (6) and fromc)andd)withj= 2(ifn= 2, we use thatMc(e) =M(e)6= 0 if and only if e∈Dn−1(C)).
Proposition 9. LetC be an abelian code in A(r1, . . . , rn). Let Γ(C)and bΓ(C)the sets of check positions defined in (8) and (13) respectively. Then Γ(C) =Γ(C).b
Proof: LetD(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 theq-orbits of D(C). TakeD a restricted set of representatives of theq-orbits of Zr1× · · · ×Zrn. Then we choose D(C) =D∩D(C)as a set of restricted representatives of theq-orbits ofD(C)and we constructΓ(C)andΓ(C).b
First, let us note that by using a)andb)in Lemma 8 we have thatΥn(C)⊆Υbn(C). Indeed, if(vn, . . . , v2)∈Υn(C)then 0≤vn≤s≤tand0≤vi≤s(vn, . . . , vi+1)≤t(vn, . . . , vi+1), withi= 2, . . . , n−1. This implies that(vn, . . . , v2)∈Υbn(C).
Now, we are going to prove thatΓ(C)⊆Γ(C). To do this we takeb (i1, . . . , in)∈Γ(C). Then there exists(vn, . . . , v2)∈Υn(C) such that
f[vn, . . . , vj]> ij ≥f[vn, . . . , vj+ 1], for every j= 2, . . . , n, and
0≤i1< g[vn, . . . , v2].
So, by the previous paragraph,(vn, . . . , v2)∈Υbn(C). By applyinga),d)ande)in Lemma 8 we have thatfb[vn, . . . , vj]> ij ≥ fb[vn, . . . , vj+ 1], for every j= 2, . . . , n, and0≤i1<bg[vn, . . . , v2]. Therefore,(i1, . . . , in)∈Γ(C).b
Let us see the inclusion bΓ(C)⊆Γ(C). Let(i1, . . . , in)∈bΓ(C). Then there exists(vn, . . . , v2)∈Υbn(C)such that fb[vn, . . . , vj]> ij ≥fb[vn, . . . , vj+ 1],
for every j= 2, . . . , n, and
0≤i1<gb[vn, . . . , v2].
We will prove that (vn, . . . , v2) ∈ Υn(C). We claim that vj 6= 0 for every j = 2, . . . , n. Assume that this is not true and let j0 be the minimum in {2, . . . , n} such that vj0 = 0. Since 0 ≤ i1 < bg[vn, . . . , v2] and bg[vn, . . . , v3,0] = 0, by the definition (12), we have that j0>2. Then, by using (11) we have thatµbvn,...,vj
0 +1,0(e) = 0, for every e∈Dj0−2, and hence t(vn, . . . , vj0+1,0) = 1. So we get the following sequence
rj0−1=fb[vn, . . . , vj0+1,0,0]>fb[vn, . . . , vj0+1,0,1]=
=fb[vn, . . . , vj0+1,0,2]= 0.
This implies thatfb[vn, . . . , vj0+1,0,1]≤ij0−1<fb[vn, . . . , vj0+1,0,0] an sovj0−1= 0. This contradicts the minimality ofj0. Then our claim is proved.
Finally we are going to show that vn 6=s+ 1 andvj 6=s(vn, . . . , vj+1) + 1, for every j = 2, . . . , n. Suppose that there exists j0 the minimum in {2, . . . , n} such that eihter vn =s+ 1, in case j0 =n, or vj0 =s(vn, . . . , vj0+1) + 1. If j0 =n then vn=s+ 1and so, by using Lemma 8 a), we have thatvn =t and0 =fb[t+ 1]=fb[t]. This is a contradiction because fb[t+ 1]≤in <fb[t]=f[s+ 1]= 0. Let us assume that j0 < n, thenvj0 =s(vn, . . . , vj0+1) + 1 =t(vn, . . . , vj0+1). Hence 0 =fb[vn, . . . , vj0+1, t(vn, . . . , vj0+1)]=f[vn, . . . , vj0+1, t(vn, . . . , vj0+1) + 1]. This contradicts that
fb[vn, . . . , t(vn, . . . , vj0+1) + 1]≤ij0<
f[vn, . . . , t(vn, . . . , vj0+1)]=fb[vn, . . . , s(vn, . . . , vj0+1) + 1]= 0.
Thus, we have proved that (vn, . . . , v2)∈Υn(C). Now, by applying a),d)ande) in Lemma 8 we have that f[vn, . . . , vj]>
ij ≥f[vn, . . . , vj+ 1], for every j = 2, . . . , n, and 0 ≤i1 < g[vn, . . . , v2]. This implies that (i1, . . . , in)∈ Γ(C) and we are done.
LetC be an abelian code inA(r1, . . . , rn)with defining set D(C)with respect to certain choice of roots of unity. Then we define
−D(C) ={(r1−e1, . . . , rn−en)|(e1, . . . , en)∈D(C)},
and we denote byC−1the 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)and−D(C)have the same q-orbits structure, and so they yield the same parametersm(·)(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 andC⊥ respectively.
The reader may find a proof in [13, p. 836].
Lemma 10. Let αe = (α1, . . . , αn) be a choice of primitive roots of unity. Let C be an abelian code in A(r1, . . . , rn) with defining set D
αe(C), and letC⊥ denote the dual code. Then D
αe(C⊥) = (Zr1× · · · ×Zrn)\D
αe(C−1).
Given C an abelian code inA(r1, . . . , rn) with defining set D(C) with respect to certain roots of unity, we denote by C0 the abelian code with defining set D(C0) = (Zr1× · · · ×Zrn)\D(C)with respect to the same choice of roots of unity. The following result follows from Lemma 10 and what we have mentioned above about C−1.
Corollary 11. Let C be an abelian code in A(r1, . . . , rn) and let C⊥ be its dual code. For a fixed set of primitive roots of unity we have that
Γ(C⊥) = Γ(C0).
The two previous results say that in order to study the relationship between bΓ(C)andΓ(Cb ⊥)we can use the setsΓ(C)b and Γ(Cb 0). The advantage of using the code C0 instead of C⊥ relies on the fact that we can compute the defining set ofC0 andC at the same time. This allows us to easily relate the respective sets of check positions.
Let us denote byfb0[·],gb0[·],Mc0(·),Ωb0(·),µb0·(·),t0andt0(·), the parameters used in the construction ofbΓ(C0). The Lemma 13 establishes the relationship between these new parameters and that used in the construction ofbΓ(C). First, we need to introduce the following recursive notation.
Notation 12. Given(un, . . . , u2)∈Υbn(C)(see (10)) we define ω1(un) =t0−un.
Suppose that we have defined ωi−1(un, . . . , un−i+2), with2≤i < n. Then we write (δ=n−i+ 2) ωi(un, . . . , uδ−1) =
[ωi−1(un, . . . , uδ), t0(ωi−1(un, . . . , uδ))−uδ−1]. We also set
ωi+(un, . . . , uδ−1) =ωi(un, . . . , ui, uδ−1−1)
= [ωi−1(un, . . . , uδ), t0(ωi−1(un, . . . , uδ))−uδ−1+ 1].
Lemma 13. LetC be an abelian code inA(r1, . . . , rn). Then, for every(un, . . . , u2)∈Υbn(C), we have that a) t=t0 yt(un, . . . , ui) =t0(ωn−i+1(un, . . . , ui)), withi= 3, . . . , n,
b) fb[un, . . . , ui]=ri−fb0[ωn−i+1+ (un, . . . , ui)], withi= 2, . . . , n, c) bg[un, . . . , u2]=r1−gb0[ωn−1(un, . . . , u2)].
Proof:Letαe= (α1, . . . , αn)be a choice of primitive roots of unity. Let us denote D(C) =D
αe(C)andD(C0) =D
αe(C0), the defining sets of C and C0 respectively, with respect to α. Lete D be a set of restricted representatives of the q-orbits of Zr1× · · · ×Zrn. Take
D(C) =D∩D(C) and D(C0) =D∩D(C0)
as sets of restricted representatives of the q-orbits of D(C) andD(C0)respectively. Then one has that D(C)∪D(C0) =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 ofD, D(C)andD(C0)are denoted by Di,Di(C)andDi(C0), respectively.
Then for everye∈Dn−1, we have thatMc(e) +Mc0(e) =rn. This implies thatt=t0 and so, by the definitions offb[·]and fb0[·], we obtainb) in casei=n.
In order to prove the remaining cases ina)andb), we use induction onn−i. The key is to prove that for everyi∈ {3, . . . , n}, (un, . . . , ui)∈Υbn−i+2(C)ande∈Di−2 the following condition holds
µbun,...,ui(e) +µb0ωn−i+1(u
n,...,ui)(e) =ri−1. (14)
Take(un, . . . , u2)∈Υbn(C). We are going to prove (14) in casei=nand consequently we will obtainb) withi=n−1 and a) with i =n. Let e∈ Dn−2. First, suppose that un 6= 0. For every a ∈R0(e), we have that a ∈R0(e)\
Ωbun(e) if and only if Mc(e, a)<fb[un]. Since we have proved b) in case i=n, we have that a∈R0(e)\
Ωbun(e)
if and only if Mc0(e, a)>fb0[ω+1(un)]. This last inequality is equivalent toa∈cΩ0ω1(un)(e)becausefb0[ω1(un)]>fb0[ω1+(un)]=fb0[ω1(un) + 1]. So,R0(e)\
Ωbun(e)
=cΩ0ω1(un)(e).
Now, suppose thatun= 0. ThenΩb0(e) =∅ andω1(0) =t=t0. By the definition of t0, we have that Mc0(e, a)≥fb0[t0]for everya∈R0(e). SoR0(e)\
Ωb0(e)
=R0(e) =cΩ0ω1(0)(e). This implies that for everyun ∈ {0, . . . , t} ande∈Dn−2 the condition (14) with i=nholds. Hence
t(un) =t0(ω1(un)) and
fb[un, un−1]+fb0[ω2+(un, un−1)]=rn−1. This gives us a)withi=nandb)withi=n−1.
Suppose that we have proved (14) for i=i0+ 1, where 2≤i0< n. Then fb[un, . . . , ui0]+fb0[ωn−i+
0+1(un, . . . , ui0)]=ri0 (15)
and
t(un, . . . , ui0+1) =t0(ωn−i0(un, . . . , ui0+1)), (16) that is, we have b)withi=i0 anda) fori=i0+ 1.
Let us prove (14) with i=i0 and we will obtainb) withi=i0−1 anda)withi=i0.
