Information sets from defining sets in abelian codes

0.37 MB·English

by  José Joaquín Bernal

1
Information sets from defining sets in abelian information sets in order to make a permutation decoding
codes attempt.
Anextendedabstractofthispaper,entitled“Informationsets
for abelian codes”, appears in the proceedings of 2010 IEEE
Jose´ Joaqu´ın Bernal and Juan Jacobo Simo´n
Information Theory Workshop, Dublin, august 30-september
3, 2010.
Abstract—We describe a technique to construct a set of check
positions (and hence an information set) for every abelian code
solely in terms of its defining set. This generalizes that given by II. PRELIMINARIES
Imai in [7] in the case of binary TDC codes. All throughout F denotes the field with q elements where q
isapowerofaprimep.LetC bealinearcodeofdimensionk,
and length l over the field F. An information set for C is a set
I. INTRODUCTION
of positions {i ,...,i } such that restricting the codewords
1 k
In the study of any family of error-correcting codes, infor- to these positions we get the whole space Fk; the other l−k
mationsetsareessentialingredientsforencodinganddecoding positions are called check positions (see [10], [11]).
1
purposes and hence it is important to describe effective algo- A (left) group code of length l is a linear code which is the
1
0 rithms for finding them. This is the interest of this paper for imageofa(left)idealofagroupalgebraviaanF-isomorphism
2 the family of abelian codes [1], [3]. FG→Fl which maps G to the standard basis of Fl. If G is a
In [7], Imai gave a method to obtain information sets for family of groups, we say that a linear code is a (left) G group
n
a binary two dimensional cyclic (TDC) codes. Later, Sakata code if it is a (left) group code for some group G belonging
J [12] gave an alternative method for the same purpose. Imai’s to G (see [2]). Many classical linear codes have been shown

0 algorithm relies in the structure of the roots of the code while to be group codes: cyclic codes, Reed-Muller codes, etc. We
1 the algorithm of Sakata is somehow based on the division deal with abelian codes, that is, G group codes where G is the

algorithm for polynomials. Up to our knowledge, these are class of finite abelian groups.
]
T the only techniques available for TDC codes. Following the We recall some basic facts about abelian codes (see
I ideas in the papers mentioned above, Chabanne described a [1], [3] for details). Every abelian code may be identi-
s. method to calculate syndromes via Groebner basis and then fied with an ideal of a group algebra FG, where G is
c he generalized the usual permutation decoding procedure (see an abelian group. It is well-known that a decomposition
[
[4]). G (cid:39) C × ··· × C , with C the cyclic group of order
r1 rn ri
1 In this paper, we generalize Imai’s method to arbitrary r , induces a canonical isomorphism of algebras FG (cid:39)
i
3v abelian codes. As it was done by Imai, our method is based F[X1,...,Xn]/(cid:104)X1r1 −1,...,Xnrn −1(cid:105).Wedenotethisquo-
on computation of cardinalities of cyclotomic cosets on tient algebra by A(r ,...,r ). Then every abelian code may
0 1 n
8 different extensions of the ground field. Such cosets, as beviewedasanidealofA(r1,...,rn),andsoweidentifythe
1 well as their cardinalities, are completely determined by codewords with polynomials P(X ,...,X ) such that every
1 n
1. the structure of the defining set of the code (see below for monomial satisfy that the degree of the variable Xi is in Zri,
0 definitions). It turns up that these cosets determine directly the cyclic group of non negative integers less or equal than
1 the shape of the information set. This relationship allows us r . We write the elements P(X ,...,X ) ∈ A(r ,...,r )
i 1 n 1 n
1 to construct codes with prescribed information sets in order as P(X ,...,X ) = (cid:80)aXj, where j = (j ,...,j ) ∈
1 n j 1 n
v: to get particular properties; or, in the case of a fixed code, Zr1 ×···×Zrn and Xj =X1j1···Xnjn.
i allows us to determine, a priori, the shape of its information We always assume that gcd(r ,q) = 1 for every i =
X i
set. This takes an advantage with respect to the use of the 1,...,n.Ourconstructionmakeuseofthestructureofrootsof
r
a Groebner basis to obtain them. As an application, we shall the ideals in A(r1,...,rn); so let us recall some basic facts
design abelian codes with good information sets in order about it. We fix a primitive r -th root of unity α in some
i i
to implement the permutation decoding algorithm (see [9]). extension of F, for each i = 1,...,n. It is well known that
Indeed, we obtain codes with better rates that those obtained everyabeliancodeC ≤A(r ,...,r )istotallydeterminedby
1 n
with other information sets. its root set, defined as
This paper is devoted to study a construction of sets of Z(C) = {(α1a1,...,αnan) | P(α1a1,...,αnan)=0
check positions for the family of abelian codes. In Section III for all P(X ,...,X )∈C}.
1 n
wedescribeamethodtoconstructinformationsetsforabelian
As it is usual, we consider the defining set of C, to witt,
codes solely in terms of their defining set. In Section IV we
include some examples which complete the exposition of our D(C) = {(a ,...,a )∈Z ×···×Z |
construction. In Section V we state our main result; to witt, 1 n r1 rn
(αa1,...,αan)∈Z(C)}.
that our set is in fact a set of check positions. In Section VI 1 n
we show how one may design abelian codes with suitable Given an abelian code C ≤ A(r ,...,r ) with defining
1 n
set D(C) with respect to {α }n , if one chooses different
DepartamentodeMatema´ticas,UniversidaddeMurcia,30100Murcia.Spain. i i=1
primitive roots of unity then the set D(C) detemines a new
email:[email protected],[email protected]
PartiallysupportedbyD.G.I.ofSpainandFundacio´nSe´necaofMurcia. code, say C(cid:48), which is permutation equivalent to C. So, for
2
the sake of shortness, we refer to abelian codes without any 1 ≤ i ≤ t < n, then γ (e) = γ (e(cid:48)) and so, by the
(cid:12) t+1 (cid:12) (cid:12)t+1 (cid:12)
mention to the primitive roots that we are using as reference. casen=1,(cid:12)C (e )(cid:12)=(cid:12)C (e(cid:48) )(cid:12)
Recallthat,forγ ∈N,theqγ-cyclotomiccosetofaninteger which implie(cid:12)s(mqγt(+π1(e),r(te+)1))=tm+1(π(cid:12) (cid:12)(e((cid:48))q)γ,t+a1s(ed(cid:48))e,rsti+re1d). t+(cid:4)1 (cid:12)
t+1 t+1
a modulo r is the set
C (a)=(cid:8)a·qγ·i | i∈N(cid:9)⊆Z . Usingthepreviousfixedorderingwebeginourconstruction
(qγ,r) r
by choosing a set of representatives of the q-orbits, which we
We extend the concept of q-cyclotomic coset modulo an
will denote by D(C), verifying the following restriction: if
integer to “q-cyclotomic coset modulo a tuple of integers”. e=(e ,...,e ) and e(cid:48) =(e(cid:48),...,e(cid:48) ) are elements of D(C)
Gitsivqe-norabnitemleomdeunlto((ar1,,......,,arn))∈asZr1 ×···×Zrn, we define such th1at γt(e)n=γt(e(cid:48)) and1et,e(cid:48)t benlong to the same qγt(e)-
1 n cyclotomic coset modulo r , for some t ∈ {1,...,n}, then
t
Q(a1,...,an) = (cid:8)(cid:0)a1·qi,...,an·qi(cid:1) | i∈N(cid:9) (1) it must happen et = e(cid:48)t. We call D(C) a set of restricted
representatives. As we will see later, every defining set D(C)
⊆ Z ×···×Z .
r1 rn hasatleastonesubsetofrestrictedrepresentatives.(Thereader
It is easy to check that may see the role of this restriction in Example 6.)
|Q(a1,...,an)|=lcm(cid:0)(cid:12)(cid:12)C(q,r1)(a1)(cid:12)(cid:12),...,(cid:12)(cid:12)C(q,rn)(an)(cid:12)(cid:12)(cid:1). wilIlnbwehfiaxtefdolaloswdse,fathuelt porredveiro.uWseoradreeridnegnoXti1ng<by··Γ·(<C)Xthne
For the sake of shortness we only write q-orbit, and the construction below [see (2)] in which we use the default
tuple of integers (r ,...,r ) will be clear by context. ordering. When we use alternative orderings we will make
1 n
It is also easy to see that D(C) is closed under multipli- adecuate notational changes (see Example 5).
cation by q in Z × ··· × Z , so, one has that D(C) is For every i ∈ {1,...,n−1}, let us denote by D (C) the
r1 rn i
necessarily a disjoint union of q-orbits. image of the projection of D(C) onto the first i-coordinates,
Conversely, every union of q-orbits modulo (r ,...,r ) and given e∈D (C), let
1 n i
defines an abelian code in A(r ,...,r ).
1 n R(e)={a∈Z |(e,a)∈D (C)},
ri+1 i+1
III. CONSTRUCTIONOFINFORMATIONSETS where (e,a) has the obvious meaning; that is, if e =
In this section we shall describe a method to construct sets (e1,...,ei) then (e,a)=(e1,...,ei,a).
of check positions for abelian codes in terms of its defining For each e∈Dn−1(C), we define
set. We assume all throughout that n > 1. We will comment (cid:88)
M(e)= m(e,a)
the case n=1 (cyclic codes) in Remark 2 (c).
Let C ≤A(r ,...,r ) be an abelian code with defining set a∈R(e)
1 n
D(C). For any element e ∈ D(C) and for each 1 ≤ i ≤ n and consider the set {M(e)} . Then we denote the
letπi(e)denotetheprojectionontothefirsti-coordinates.We different values of the M(e)’se∈aDsnf−o1ll(oCw) s,
set D (C) = π (D(C)); moreover, for each 1 ≤ t ≤ i and
i i
e ∈ D (C) we also denote by π (e) its projection onto the f[1] = max {M(e)} and
i t
first t-coordinates. e∈Dn−1(C)
We fix a well-ordering in the indeterminates setting X < f[i] = max {M(e)|M(e)<f[i−1]}.
···<X ,andwedefinerecursivelythefollowingparamet1ers. e∈Dn−1(C)
n
For each e = (e ,...,e ) ∈ D (C), with 1 ≤ j ≤ n, we set So, we obtain the sequence
1 j j
(cid:12) (cid:12)
m1≤(πt1(≤e)j)−=1(cid:12),Cw(qe,rs1e)t(e1)(cid:12), and having m(πt(e)) defined for f[1]>···>f[sn]>0=f[sn+1],
t where f[sn+1]=0 by convention.
(cid:89)
γ (e)= m(π (e)) Thisistheinitialsequenceforanyvalueofn.Now,suppose
t+1 l
l=1 that n≥3. Then we continue as follows:
and Given 1≤un ≤sn, we define for every e∈Dn−2(C)
(cid:12) (cid:12)
m(π (e))=(cid:12)C (e )(cid:12).
t+1 (cid:12) (qγt+1(e),rt+1) t+1 (cid:12) Ωun(e)={a∈R(e)|M(e,a)≥f[un]}
Finally, for e ∈ D(C), we set γ (e) = (cid:81)n m(π (e)).
n+1 l=1 l and
It is easy to see that for 1 ≤ t ≤ n, we have that (cid:88)
µ (e)= m(e,a).
γ (e)=|Q(π (e))|andfor1≤t<n,onehasm(π (e))= un
t+1 t t
γ (e)/γ (e). a∈Ωun(e)
t+1 t
Note that the set Ω (e) may eventually be the empty set. In
Remark 1. Given a q-orbit, any of its representatives yields un
this case, the corresponding value µ (e) will be zero.
the same parameters m’s and γ’s. un
We define
Let us prove this assertion. We argument by induction.
fTifohrCea(ncqay,rs)eγ(an∈) ==N.1CSu(iqps,pr)toh(saee(cid:48))othbtahvteionmus(cid:12)(cid:12)(Cπs(tqa(γete,)rm))(e=an)t(cid:12)(cid:12):mf=(oπr(cid:12)(cid:12)(aCe,((cid:48)aq)γ)(cid:48),rf∈)o(raZ(cid:48)a)rl(cid:12)(cid:12)l, ff[[uunn,,1i]] == e∈Dmmnaa−xx2(C){{µµuunn((ee))}|0<µaunnd(e)<f[un,i−1]}.
i i e∈Dn−2(C)
3
For each 1 ≤ u ≤ s we order the previous parameters The main result of this paper stays that Γ(C) is a set of
n n
getting the sequence check positions for C (see Theorem 9) or equivalently, the
complement of Γ(C) in Z ×···×Z is an information set
f[un,1]>···>f[un,s(un)]>0=f[un,s(un)+1], of C. r1 rn
where, again, f[un,s(un)+1]=0 by convention. Remarks 2. a) In the case n=2 the reader may check that
Now, suppose that n≥j ≥4 and we have constructed for
the set
each list (u ,...,u ), where 1 ≤ u ≤ s and 1 ≤ u ≤
n j n n l
s(un,...,ul+1), for j ≤l<n, the sequence Γ(C) = {(i1,i2)∈Zr1 ×Zr2 |∃u∈{1,...,sn}
with (3)
f[un,...,uj,1]>···>f[un,...,uj,s(un,...,uj)]>0= f[u+1]≤i2 <f[u] and 0≤i1 <g[u]}
=f[un,...,uj,s(un,...,uj)+1]. is that constructed by Imai for TDC codes in [7], viewed
under our notation.
Then we define for each e ∈ D (C) and 1 ≤ u ≤
j−3 j−1 b) The construction of Γ(C) depends uniquely on the values
s(u ,...,u )
n j m(−) computed over the restricted sets of representatives,
(cid:8)
Ω (e) = a∈R(e)|µ (e,a)≥ and, as we have seen in Remark 1, this values are inde-
un,...,uj−1 un,...,uj
f[un,...,uj,uj−1]} pendent of the representatives, provided that their election
respects the restriction imposed. Therefore, any election
and of restricted representatives D(C) yields the same set of
(cid:88)
µun,...,uj−1(e)= m(e,a), check positions as far as the order of the indeterminates is
a∈Ωun,...,uj−1(e) fixed.However,thevaluesm(−)dependontheorderingof
theindeterminates,sothatdifferentorderingsmayproduce
and by ordering these parameters we obtain the sequence
differentsetsofcheckpositions,asitisshowninExample
f[un,...,uj−1,1] > ···>f[un,...,uj−1,s(un,...,uj−1)] 5.
> 0=f[un,...,uj−1,s(un,...,uj−1)+1]. c) Cyclic codes. Let C be a cyclic code of length l
over F with root set Z(C) and defining set D(C) =
We continue in this way until we have defined, for each {e∈Z | αe ∈Z(C)},whereα isaprimitivel-throotof
l
(u ,...,u ), the sequence
n 3 unity. Assume that gcd(l,q)=1. Let D(C) be a complete
set of representatives of the q-cyclotomic cosets modulo
f[un,...,u3,1]>···>f[un,...,u3,s(un,...,u3)]>0=
l of the elements of D(C). The specialization for n = 1
=f[un,...,u3,s(un,...,u3)+1]. of the construction given above yields the set of check
Finally, for any value of n, we define the last list of param- positions
etersasfollows:foreverylist(u ,...,u ),with1≤u ≤s (cid:88)
n 2 n n Γ(C)={i ∈Z |0≤i < m(e)};
and 1 ≤ u ≤ s(u ,...,u ), with 2 ≤ l < n, if n = 2 we 1 l 1
l n l+1
set e∈D(C)
(cid:88)
g[u2]= m(e) that is, the last l−|D(C)| positions define an information
set, as it is well known (see [10]).
e∈D1(C)
M(e)≥f[u2] Now suppose that l = r ·r with gcd(r ,r ) = 1. Then
1 2 1 2
and, in case 2<n, we set there exists a group isomorphism ϕ : C → C × C
l r1 r2
(cid:88) whichcanbeextendedbylinearitytoanisomorphismϕof
g[un,...,u2]= m(e). groupalgebrasfromFC toF(C ×C ).Moreover,fixed
µun,...,ue3∈(eD)≥1(fC[)un,...,u2] α1, α2 primitive r1-th,lr2-th roort1s of urn2ity, ϕ maps D(C)
onto the set D(ϕ(C)). So, we can compute an information
Usingallthesequencesobtainedpreviously,wemaydefine
set of ϕ(C) and then take the inverse image of those
the set
positions by ϕ, obtaining a new information set for C.
(cid:40) n It is not hard to see that different isomorphisms may
Γ(C) = (i ,...,i )∈(cid:89)Z |∃(u ,...,u ) produce different sets of check positions for C. Indeed, if
1 n ri n 2 ϕ,ϕ(cid:48) : C → C ×C are different group isomorphisms
i=1 l r1 r2
where (2) thenbeingD(ϕ(C))andD(ϕ(cid:48)(C))differentornot,thesets
of check positions coincide; that is Γ(ϕ(C))=Γ(ϕ(cid:48)(C)).
1≤u ≤s , and
n n This is because group isomorphisms preserve the structure
1≤ul ≤s(un,...,ul+1), for l=2,...,n−1 of cyclotomic cosets (and so the values m(−)). Thus, as
such that ϕ (cid:54)= ϕ(cid:48), their inverses may produce different information
f[un+1]≤in <f[un], sets for C.
Let us give briefly an example. Let C be the binary cyclic
···
code of length 15 with root set Z(C) = {αe | e =
f[un,...,u2+1]≤i2 <f[un,...,u2], 0,1,2,3,4,6,8,9,12} where α is a primitive 15-th root
0≤i1 <g[un,...,u2]}. ofunity.UsingtheChineseRemainderTheoremwedefine
4
the group isomorphism ϕ : C → C × C given by Take D (C) = {0,1} a complete set of representatives
15 3 5 1
ϕ(t) = (t ,t ), where t ≡ t mod 3 and t ≡ t mod 5 of the 2-cyclotomic cosets modulo 3 of the elements of
1 2 1 2
are suitable representatives. D (C). Then we may take D (0) = {(0,3)} and D (1) =
1 2 2
Then,consideringα andα primitivethirdandfifthroots {(1,1),(1,3)}. By definition, D(C )={(0,3),(1,1),(1,3)}.
1 2 1
of unity, respectively, such that α=α α , we obtain Adirectcomputationyieldsm(0)=1,m(0,3)=m(1,1)=
1 2
m(1,3) = 3, and m(1) = 2. Hence M(0) = 3 and
Z(ϕ(C)) = {(1,1),(1,α ),(1,α2),(1,α3),(1,α4),
2 2 2 2 M(1)=6. Then, we obtain the (unique) sequence
(α ,α ),(α2,α2),(α ,α4),(α2,α3)}
1 2 1 2 1 2 1 2
f[1]=6>f[2]=3>0=f[3]
and
and
D(ϕ(C)) = {(0,0),(0,1),(0,2),(0,3),(0,4),(1,1),
g[1]=2<g[2]=3.
(2,2),(1,4),(2,3)}.
Finally, according to (3) the set of check positions for C is
1
Using our construction we get the set of check positions
Γ(C ) = {(0,0),(1,0),(2,0),(0,1),(0,2),(0,3),
for ϕ(C), 1
(0,4),(0,5),(1,1),(2,1),(1,2),(2,2),
Γ(ϕ(C)) = {(0,0),(0,1),(0,2),(0,3),(0,4),(1,0),
(1,3),(1,4),(1,5)}
(2,0),(1,1),(2,1)}.
Going back we obtain the set of check positions for C,
ϕ−1(Γ(ϕ(C)))={0,1,3,5,6,9,10,11,12};
a non obvious one. The reader may check that the isomor-
phismϕ(cid:48)(t)=(3−t ,t ),witht ,t asabove,determines
1 2 1 2
another different set of check positions for C.
It is easy to see that this can be generalized to the case
l=r ···r with pairwise coprime r ’s.
1 n i
IV. EXAMPLES
BeforeprovingthefactthatΓ(C)isasetofcheckpositions
of C, we present some examples to illustrate our construction.
LetC beanabeliancodewithdefiningsetD(C).Toconstruct
a set of check positions we first need a set of restricted
representatives of the q-orbits of D(C); that is, D(C). We
propose the following inductive algorithm using the default
ordering X <···<X .
1 n
• Takeacompletesetofrepresentativesoftheq-cyclotomic
cosetsmodulo r ofthe elementsof D (C).This setwill
1 1 Fig.1. Γ(C1)
beD (C),andforthesakeofsimplicityweusethesame
1
symbol.
Example 4. Set q = 2,n = 3,r = r = r = 3. Let C be
• Suppose that we have defined Di−1(C). Given e ∈ 1 2 3 2
the abelian code with defining set
D (C) we define D (e) as the set of the elements
i−1 i
(e,ai) where ai runs through a complete set of repre- D(C2) = {(0,0,0),(1,1,0),(2,2,0),(0,1,1),(0,2,2),
sentatives of the qγi(e)-cyclotomic cosets modulo ri of (2,2,1),(1,1,2)}.
the elements of D (e) = {a ∈ Z | (e,a) ∈ D (C)}.
i ri i
One should have in mind that if e,e(cid:48) ∈ Di−1(C) and Take D1(C) = {0,1} as a complete set of representatives
γ (e)=γ (e(cid:48))thenwemusttakethesamerepresentatives of the 2-cyclotomic cosets modulo 3 of the elements of
i i
for each qγi(e)-cyclotomic coset in Di(e)∩Di(e(cid:48)). Set D1(C). Choose D2(0) = {(0,0),(0,1)}, D2(1) = {(1,1)}.
Di(C)= (cid:83) Di(e). By definition, D2(C) = {(0,0),(0,1),(1,1)}. Now, we
e∈Di−1(C) set D3(0,0) = {(0,0,0)}, D3(0,1) = {(0,1,1)}, and
• Finally, we define D(C)=Dn(C). D3(1,1)={(1,1,0),(1,1,2)}. Then
Clearly, any set of restricted representatives, D(C), comes
D(C )={(0,0,0),(0,1,1),(1,1,0),(1,1,2)}.
2
from elections in the given way.
A direct computation shows that m(0) = m(0,0) =
Example 3. Let q = 2, n = 2, r = 3, r = 7, and consider
1 2 m(1,1) = m(0,0,0) = m(0,1,1) = m(1,1,0) =
the abelian code C with defining set
1 m(1,1,2) = 1, and m(1) = m(0,1) = 2. Therefore
D(C ) = {(1,1),(2,2),(1,4),(2,1),(1,2),(2,4), M(0,0) = M(0,1) = 1,M(1,1) = 2 which, in turn, give
1
(0,3),(0,5),(0,6),(1,3),(2,6),(1,5), us the first sequence
(2,3),(1,6),(2,5)}. f[1]=2>f[2]=1>0=f[3].
5
Foru =1wehavethevaluesµ (0)=0,µ (1)=1.This Now we choose the other order, X <X . We take
3 1 1 2 1
yields a sequence
D(C )={(0,0),(1,0),(2,1)}.
3
f[1,1]=1>0=f[1,2].
Then m(0) = 1, m(1) = 4, m(0,0) = 1, m(1,0) = 2,
For u3 =2 the values are µ2(0)=3,µ2(1)=1. Then, m(2,1) = 1. Using this, we have M(0) = 3, M(1) = 1. So
we obtain the sequence
f[2,1]=3>f[2,2]=1>0=f[2,3].
and finally f[1]=3>f[2]=1>f[3]=0
g[1,1]=2 and g[2,1]=1<g[2,2]=3. and
g[1]=1<g[2]=5.
Following (2), the set of check positions for C is
Therefore, the set of check positions for C with respect to
3
Γ(C ) = {(0,0,0),(1,0,0),(2,0,0),(0,1,0),
2 the order X <X is
2 1
(0,2,0),(0,0,1),(1,0,1)}.
Γ(C ;X <X ) = {(0,0),(1,0),(2,0),(0,1),(0,2),
3 2 1
(0,3),(0,4)}.
Our last example shows that the restriction on the election
of representatives is not superfluous, and allows us to avoid
redundance on the sums.
Example 6. Set q = 2,n = 3,r = r = r = 3. Let C be
1 2 3 4
the abelian code with defining set
D(C )={(0,0,0),(0,1,1),(0,2,2),(0,2,1),(0,1,2)}.
4
One may choose D(C ) = {(0,0,0),(0,1,1),(0,1,2)} or,
4
alternatively, D(C ) = {(0,0,0),(0,2,1),(0,2,2)}; but we
4
must not select as D(C ) the set {(0,0,0),(0,1,1),(0,2,1)}.
4
If we do this last election, we will obtain µ (0)=m(0,0)+
1
m(0,1)+m(0,2)=5,whichmakesnosense,becauser =3,
i
for i=1,2,3. Note that the election of (0,1) and (0,2) as an
D (C) does not respect the rules of restricted representatives.
2
V. MAINTHEOREM
Fig.2. Γ(C2)
In this section we will prove that Γ(C) is a set of check
positions. Firstly, we need to introduce the notion of check
tensor associatedwithanabeliancode C.Thisdefinitionis an
Next example shows that different linear orderings in the
extension to n indeterminates of that given by Imai in [7].
indeterminates may yield different sets of check positions for Let C ≤A(r ,...,r ) be an abelian code with defining set
1 n
an abelian code.
D(C). Suppose we have ordered X < ... < X obtaining
1 n
Example 5. Let q =2, n=2, r =3 and r =5. Let C be D(C) and Γ(C) as it was done in Section III.
1 2 3
the abelian goup code with defining set Let Fq(cid:48) denote the field of q(cid:48) elements with q(cid:48) a power of
q (recall from the Preliminaries that F = F ). For every j =
q
D(C3)={(0,0),(1,0),(2,0),(1,2),(2,4),(1,3),(2,1)}. (j ,...,j )∈Z ×···×Z , and every e=(e ,...,e )∈
1 n r1 rn 1 n
First we choose the (default) order X <X . We take D(C)wedefinehj astheexpressionofαe1·j1·····αen·jn ina
1 2 e 1 n
D(C )={(0,0),(1,0),(1,2)}. fixedbasisoftheFq-vectorspaceFq(cid:48),whereq(cid:48) =qγn+1(e)and
3 α is an r -th primitive root of unity, for every i = 1,...,n.
i i
Then m(0) = 1, m(1) = 2, m(0,0) = 1, m(1,0) = 1, Now, we fix an arbitrary order in D(C) and following it, for
m(1,2) = 2. Using this, we have M(0) = 1, M(1) = 3. So each j ∈ Z ×···×Z , we concatenate the vectors hj to
we obtain the sequence get a new vre1ctor h =(cid:0)rhnj(cid:1) . e
j e e∈D(C)
Observe that, the equality
f[1]=3>f[2]=1>f[3]=0
(cid:88) (cid:88)
γ (e)= |Q(e)|=|Z(C)|
and n+1
g[1]=2<g[2]=3. e∈D(C) e∈D(C)
impliesthatthelengthofh consideredasavectorwithentries
Therefore, the set of check positions for C with respect to j
3 in F is |Z(C)|. This means that the F -subspace spanned
this order is q q
by the h ’s verifies (cid:104)h | j∈Z ×···×Z (cid:105) ⊆ F .
Γ(C ;X <X ) = {(0,0),(1,0),(2,0),(0,1),(0,2), j j r1 rn q|Z(C)|
3 1 2 In fact, as a consequence of Corollary 10, we will see that
(1,1),(1,2)}. equality holds.
6
Definition 7. Let C ≤ A(r ,...,r ) be an abelian code. A 2,...,n−1
1 n
check tensor for C is a set of the form
s(un,(cid:88)...,ui+1)
σi(un,...,ui+1)= (f[un,...,ui+1,ui]−
{hj | j∈Zr1 ×···×Zrn}, ui=1
s(un(cid:88),...,u3)
where h =(cid:0)hj(cid:1) ∈F and, in turn, D(C) is a set f[un,...,ui+1,ui+1])··· (f[un,...,u3,u2]−
of restricjted reepree∈seDn(tCa)tivesqo|Zf(tCh)e| q-orbits of D(C). u2=1
f[un,...,u3,u2+1])·g[un,...,u2].
Note that by definition, P(X ,...,X )=(cid:80)λ Xj1·
z·e·r·o·Xvenjncto∈rCwiifthanledngotnhly|Zif((cid:80)C)|j.1λjhj =0n, where 0jd1,e..n.,ojtnes1the Then, it is clear(cid:88)tshnat
|Γ(C)|= (f[un]−f[un+1])σn−1(un)
Lemma8. AsetH⊆Z ×···Z isasetofcheckpositions un=1
r1 rn
for C if and only if the set {h } is a basis of the subspace and
j j∈H
generated by the check tensor. s(un,(cid:88)...,ui+2)
Proof: First note that H is a set of check positions if and σi+1(un,...,ui+2)= (f[un,...,ui+2,ui+1]− (4)
only if any codeword is uniquely determined by its entries ui+1=1
in the information set Hc, the complementary set of H in f[un,...,ui+2,ui+1+1])·σi(un,...,ui+1).
Zr1 ×···×Zrn. We claim that for i=2,...,n−1 one has
Assume that H is a set of check positions. Suppose that (cid:88)
thereexistcoefficients{λ } ⊆F satisfying(cid:80) λ h = σi(un,...,ui+1)= µun,...,ui+1(e)·γi(e). (5)
j j∈H q j∈H j j
0. Define c = (c ) such that c = 0 if j ∈/ H and c = λ if e∈Di−1(C)
j j j j
j ∈ H. If one of the coefficients of the linear combination is We proceed by induction. Taking i=2 as induction base,
not zero, one has that c and the zero codeword are different
elements of C and they agree in the positions corresponding s(un(cid:88),...,u3)
to Hc, which is impossible. This proves linear independence. σ2(un,...,u3)= (f[un,...,u3,u2]−
Now, pick j ∈/ H. Then, there exists {λ } ⊆F such that u2=1
0 j j∈H q f[un,...,u3,u2+1])·g[un,...,u2].
the vector c=(c ), defined by
j
For every e ∈ D (C), set µ(e) = µ (e). Having in
1 un,...,u3
 mind that
0 if j∈Hc\{j },
 0
(cid:88)
cj = 1 if j=j0, g[un,...,u2]= m(e),
 λ if j∈H
j e∈D1(C)
µ(e)≥f[un,...,u2]
(cid:80) we factor out m(e) obtaining
is a codeword of C. So, h + λ h =0.
j0 j∈H j j
(cid:80)
Conversely, suppose that {h } is linearly independent σ (u ,...,u )= m(e)·
j j∈H 2 n 3
aFnqd-lifnoerarevceormybiin∈atiZorn1o×f t·h·e·×eleZmrenn,tshiofca{nhjb}ej∈wHr.itten as an (cid:80) (ef∈[Dun1(,C.)..,u3,u2]−f[un,...,u3,u2+1])
poTthaekseis,antyhecovdeecwtoorrd(cid:80)c = (ccjh)j∈Zhra1s×·a··×uZnrniquoef Cex.pBreysshioyn- =f[un,.(cid:80)..,u3,um2]≤(eµ)(e·)µ(e)= (cid:80) γ2(e)·µun,...,u3(e).
(cid:80) c h = (cid:80) λ hj∈,Hfcorjsojme λ ∈ F . As c ∈ C then e∈D1(C) e∈D1(C)
j∈Hc j j j∈H j j j q So the induction base is stablished. Now, we assume that
λ = −c , for all j ∈ H, which means that the codeword is
j j the claim has been proved for i<n−1 and we are going to
totally determined by those {c } .
j j∈Hc deal with the case i+1. By (4) and by assumption
Now we state our main theorem. s(un,(cid:80)...,ui+2)
σi+1(un,...,ui+2)= (f[un,...,ui+1]−
dTehfienoirnegmse9t.DL(eCt)C. T≤heAn(rΓ1(,C.)..i,sran)sebteoafnchaebcekliapnosciotidoenswfiothr f[un,...,ui+1+1])· (cid:80)ui+1=µ1un,...,ui+1(e)·γi(e)
C. e∈Di−1(C)
s(un,(cid:80)...,ui+2)
= (f[un,...,ui+1]−f[un,...,ui+1+1])·
{h }Proof:iBsya Lbaesmismoaf8theitFis-veencotourghsptaoceprgoevneertahteadt tbhye tsheet ui+1=1 (cid:32) (cid:33)
j j∈Γ(C) (cid:80) (cid:80)
γ (e)· m(e,a) .
check tensor; furthermore, we are going to show that it is a i
basis of the whole space F . e∈Di−1(C) µun,...,ui+2(e,a)≥f[un,...,ui+1]
q|Z(C)|
First, we will prove that |Γ(C)| = |Z(C)|. For any Note that given e = (e ,...,e ) ∈ D (C), with
1 i i
fixed (un,...,u2), with 1 ≤ un ≤ sn and 1 ≤ ul ≤ µun,...,ui+2(e) = f[un,...,ui+2,l] for some 1 ≤ l ≤ n,
s(u ,...,u ), with 2 ≤ l < n, let us define for i = one has that π (e) ∈ D (C). Moreover, e satisfies
n l+1 i−1 i−1 i
7
that µun,...,ui+2(e1,...,ei−1,ei) ≥ f[un,...,ui+2,ui+1] for and while l<n,
all u ≥ l. Conversely, for e ∈ D (C) and a ∈
i+1 i−1
R(e), there must exist u such that µ (e,a) ≥
fha[uvne,...,ui+2,ui+1]. Then,i+se1tting µ(e) = µunu,n..,..,.u.,iu+i2+2(e), we µvn,...,vl+2(e1M,..(.e,1e,l.).≥.,efn[v−n1,)..≥.,vfl[+v1n]] iiff ll≤=nn−−21,. ((87))
σi+1(un,...,ui+2)= (cid:80) γi+1(e)· We begin with l = 1. By the definition of v¯i and since
(cid:80) e∈Di(C) i ∈ Γ(C) we have that 1 ≤ i1 < g[vn,...,v2]. Consider the
(f[un,...,ui+1]−f[un,...,ui+1+1]) polynomial
=f[un,.(cid:80)..,ui+1γ]≤i+µ1((ee))·µ(e). P(X1)= (cid:88) λjX1j1.
e∈Di(C) j=(j1,i2,...,in)∈Γ(C)
So (5) is proved. Now, using (5) with i = n−1, and the By definition of Γ(C), we have u¯ = v¯, for j =
j i
definitions of µun(e) and M(e), we deduce (j1,i2,...,in) ∈ Γ(C) and then j1 must verify 1 ≤
(cid:80)sn j1 < g[vn,...,v2]. Conversely, for any 1 ≤ j1 <
|Γ(C)| = (f[un]−f[un+1])· g[vn,...,v2] one has (j1,i2,...,in) ∈ Γ(C). So δ(P(X1)) =
un=(cid:80)1 γ (e)·µ (e) |{j1 | 1≤j1 <g[vn,...,v2]}|<g[vn,...,v2],whereδ denote
n−1 un the polynomial degree.
e∈Dn−2(C)
(cid:80)sn (cid:80) Now, since
= un=1(f(cid:80)[un]−f[un+1])·e∈Dn−2(C)γn−1(e) g[vn,...,v2] = (cid:12)(cid:12)(cid:12)(cid:91)(cid:8)C(q,r1)(a) | a∈D1(C) and
· m(e,a)
M(e,a)≥f[un] µvn,...,v3(a)≥f[vn,...,v2]}|
(cid:80) (cid:80)
= γn(e)· (f[un]−f[un+1]) and P(X1)∈Fq[X1] then there exists e1 ∈D1(C) satisfying
= e∈D(cid:80)n−1(C)γ (e)·fM[u(ne])≤=M(e)(cid:80) γ (e). (7) for l=1, and also P(α1e1)(cid:54)=0. This proves the induction
n n+1 step l=1.
e∈Dn−1(C) e∈D(C) Suppose we have constructed e = (e ,...,e ) ∈ D (C)
1 l l
Thisequality,togetherwiththedefinitionofγn+1(e)yields satisfying (6) and (7). We want to get the step l+1.
(cid:88) (cid:88) Consider the polynomial
|Γ(C)|= γ (e)= |Q(e ,...,e )|=|Z(C)|,
n+1 1 n
P (X )=
e∈D(C) e∈D(C) e l+1
(cid:88) λ ·αe1j1···αeljl ·Xjl+1.
and we are done. j 1 l l+1
Now,wehavetoseethat{h } islinearlyindependent. j=(j1,...,jl+1,il+2,...,in)∈Γ(C)
j j∈Γ(C)
So,considerafamily{λ } ofcoefficientsinF,inwhich Then, for any j = (j ,...,j ,i ,...,i ) ∈ Γ(C), we
j j∈Γ(C) 1 l+1 l+2 n
thereisatleastoneλj (cid:54)=0,forsomej∈Γ(C).Weshallshow have that u¯j = (un,...,u2) verifies un = vn,...,ul+2 =
that (cid:80)j∈Γ(C)λjhj (cid:54)=0, and this will finish the proof. vl+2.Bydefinitionof v¯i wehavethat λj (cid:54)=0implies u¯j ≥v¯i;
Let us define the set which, in turn, implies f[vn,...,vl+1]≥f[vn,...,ul+1]. Thus,
Υ = {(u ,...,u ) | 1≤u ≤s , and f[vn,...,ul+1+1] ≤ jl+1 < f[vn,...,ul+1] ≤ f[vn,...,vl+1].
n 2 n n Then δ(Pe(Xl+1)) < f[vn,...,vl+1] and, by induction hy-
1≤ul ≤s(un,...,ul+1), for 2≤l≤n−1} pothesis, we have f[vn,...,vl+1]≤µvn,...,vl+2(e).
Analogously to the case l=1 we have that, since
ordered in the usual lexicographical order: (u ,...,u ) ≤
n 2
(aZund(cid:48)n,×.u.i·.·,≤·u×(cid:48)2)uZi(cid:48)if.uRnebc≤ealoluln(cid:48)ngthsoarttoifaΓunj(Ce=l)emuif(cid:48)jeanfnotdrjjo==nlyn(,ijf.1.,t.h.,.ei.r+e,j1enx)≥ist∈2s µvDn,l.+..1,v(lC+2)(aen)d=µ(cid:12)(cid:12)(cid:12)(cid:83)vn,(cid:110)...C,v(lq+γ3l+(e1,(ea)),rl≥+1f)([avn),.|..(,ev,l+a2)](cid:9)∈(cid:12)(cid:12)
aru1nique (un,r.n..,u2) ∈ Υ, which we call u¯j, such that and Pe(Xl+1) ∈ Fqγl+1(e)[Xl+1], then there exists el+1 ∈
f[un,...,ul+1] ≤ jl < f[un,...,ul] for l = 2,...,n, and Zrl+1 satisfying(6)and(7).Nowwemaygetthestepl=n−1
1≤j1 <g[un,...,u2]. in a complete analogous way obtaining (8), instead of (7).
We have to prove that there is an element (e1,...,en) ∈ Finally, suppose we have an element e=(e1,...,en−1)∈
D(C) such that Dn−1(C), satisfying (6), (7) and (8). Consider the polynomial
(cid:88) λj·α1e1j1···αnenjn (cid:54)=0. Pe(Xn)= (cid:88) λj·α1e1j1···αlen−1jn−1 ·Xnjn.
(j1,...,jn)∈Γ(C) j=(j1,...,jn)∈Γ(C)
To do this, we begin by fixing v¯i = As in the previous steps, if λj (cid:54)= 0 then u¯j ≥ u¯i;
min{u¯j ∈Υ | λj (cid:54)=0}; where i = (i1,...,in) ∈ Γ(C). Set hence un ≥ vn. This means that f[un] ≤ f[vn], and since
(v¯ei1=,..(.v,ne,n.)..∈,vD2)(.CW) seuschhatlhlactofnosrtreuaccthrelc=urs1i,v.e.ly.,ann, element ff[[vunn]+≤1] M≤(ej)n=<(cid:80)fa∈[uRn(]e)≤m(ef,[avn).],Twhues,htahveereδis(aPne(eXnn∈))Zr<n
such that (e1,...,en)∈D(C) and Pe(αnen)(cid:54)=0; that is
(cid:88) λj·α1e1j1···αleljl (cid:54)=0 (6) (cid:88) λj·α1e1j1···αnenjn (cid:54)=0
j=(j1,...,jl,il+1,...,in)∈Γ(C) j=(j1,...,jn)∈Γ(C)
8
as desired. the information symbols of the permuted vector τ(r) are
correct, so by using the parity check equations we obtain the
Thefollowingcorollaryiswell-known(seeforinstance[6]).
redundancy symbols and then we can construct a codeword
Corollary 10. Let C be an abelian code with defining set c(cid:48). Finally, we decode to τ−1(c(cid:48))=c.
D(C). Suppose C has dimension k and length l. Then k = In general to find t-PD-sets for a given t-error correcting
l−|Z(C)|. code is not at all an easy problem. It depends on the chosen
information set and they need not even to exist. Moreover, it
VI. APPLICATIONSTOPERMUTATIONDECODING isclearthatthealgorithmismoreefficientthesmallerthesize
of the PD-set.
In this section we present some ideas on how one may de-
Let T be the transformation from A(r ,...,r ) into itself,
signabeliancodeswithgoodinformationsetsinordertomake s 1 n
given by T (P(X ,...,X ))=X ·P(X ,...,X ), for s=
a permutation decoding attempt. By using those information s 1 n s 1 n
1,...,n. Then it is clear that T can be seen as a permutation
sets obtained in Section III, we present two examples. The s
in S , via T (i ,...,i ) = (i ,...,i +1,...,i )
first example has two parts. In the first one, we give a list r1×···×rn s 1 n 1 s n
and as such, (cid:104){T }n (cid:105) may be viewed as a subgroup of
of 2-error correcting cyclic codes of length 45 that achieve s s=1
PAut(C) for every abelian code C ≤ A(r ,...,r ). On the
the upper bound on the ranks that appears in [13]. Next, we 1 n
other hand, we consider the Frobenius automorphism of F=
construct two 2-error correcting abelian non cyclic codes of (cid:16) (cid:17)
F acting on A(r ,...,r ) via σ (cid:80) a Xj = (cid:80) a Xq·j
length 45 with better rates than those given for cyclic ones. q 1 n j j j j
The other example is a list of 3-error correcting cyclic codes (recall that gcd(ri,q) = 1, for all i = 1,...,n). As well as
of length 65 that improve the upper bound on the ranks that those Ts’s, the Frobenius automorphism belongs to PAut(C),
appears in [14]. for every abelian code C ≤ A(r1,...,rn). We shall look for
Before we present our examples let us give a brief intro- PD-sets contained in the subgroup Λ=(cid:104){Ts}ns=1∪{σ}(cid:105).
duction to the permutation decoding algorithm. Permutation It is well-known that for every cyclic code of dimension k,
decoding was introduced by F. J. MacWilliams in [9] and it anykconsecutivepositionsformaninformationset(see[10]).
is fully described in [5] and [10]. Fixed an information set In [13], [14] Shiva, Fung and Tan gave upper bounds on the
for a given linear code C, this technique uses a special set of rank of cyclic codes of certain lengths, that are permutation
permutation automorphisms of the code called PD-set. decodablewithrespecttotheinformationsetmentionedabove
We denote the permutation group on Z × ··· × Z and with PD-sets contained in Λ. For our purposes, we shall
r1 rn
by S and we consider its extension to automor- interpret cyclic codes as abelian codes (in several variables),
r1×···×rn (cid:16) (cid:17)
phisms of A(r ,...,r ) via τ (cid:80) aXj = (cid:80) a Xj. asitwasdoneinRemarks2(c).Thenwedesignabeliancodes
1 n j j j τ−1(j)
such that Λ contains a PD-set with respect to the information
Under this point of view the permutation automorphism
group of an abelian code C ≤ A(r ,...,r ) is PAut(C) = set defined in (2) and their rates achieve and even improve
1 n
those given by Shiva, Fung and Tan.
{τ ∈S | τ(C)=C}.
r1×···×rn We only consider abelian codes in two variables; however
Definition 11. Let C be a t-error-correcting [l,k] code. Let I the procedure may be extended to more variables. So, in the
be an information set for C. For s≤t a s-PD-set for C and I sequel, C ≤ A(r ,r ) will be an abelian code with set of
1 2
is a subset P ⊆PAut(C) such that every set of s coordinate check positions Γ(C) given by (3), and Q(e ,e ) will denote
1 2
positions is moved out of I by at least one element of P (see the q-orbit of (e ,e )∈Z ×Z as in (1).
1 2 r1 r2
[8], [9]).
We begin by dealing with 2-PD-sets. In this case, if Γ(C)
intersectsallq-orbitsinZ ×Z ,wemaymovethesupportof
The idea of permutation decoding is to apply the elements r1 r2
an 2-error vector into Γ(C) by using the subgroup Λ. Indeed,
ofthePD-settothereceivedvectoruntiltheerrorsaremoved
take e∈A(r ,r ) an error vector with supp(e)={p ,p }⊆
outofthefixedinformationset.Thefollowingtheoremshows 1 2 1 2
Z ×Z andsetp =(p1,p2)forj =1,2.Thenuse(cid:104)T ,T (cid:105)
howtocheckthattheinformationsymbolsofacodewordwith r1 r2 j j j 1 2
weight less or equal than t are correct. We denote the weight totransformp1 (cid:55)→(0,0)andp2 (cid:55)→p,forsomep∈Zr1×Zr2.
of a vector v ∈Fl with wt(v). Finally,asthereexistsanintegerisuchthatqi·p∈Γ(C),then
σi(0,0)=(0,0) and σi(p) belong to Γ(C). Summarizing
Theorem12([5],Theorem8.1). LetC beat-error-correcting
[l,k] code with parity check matrix H in standard form. Lemma 13. Let C ≤A(r1,r2) be an abelian t-error correct-
Let r = c + e be a vector, where c ∈ C and wt(e) ≤ t. ing code with set of check positions Γ(C). If t ≥ 2 and Γ(C)
Then the information symbols in r are correct if and only if intersects all q-orbits in Zr1×Zr2, then Λ is a 2-PD-set with
wt(cid:0)HrT(cid:1)≤t. respect to Γ(C)c.
Then, once we have found a PD-set P ⊆ PAut(C) for Example 14. We shall design binary 2-error correcting codes
the given code C and information set I, the algorithm of with the highest dimension in length 45, such that Λ contains
permutation decoding is as follows: take a check-matrix H a 2-PD-set with respect to our information set.
for C in standard form. Suppose that we receive a vector Firts of all, let us note that it is possible to check, by using
r = c+e with wt(e) ≤ t. Then we calculate the syndromes the GAP program, that any abelian code (cyclic or not), C, of
H(τ(r))T, with τ ∈ P, until we obtain a vector H(τ (r))T length45withdim(C)≥33hasminimumdistanced(C)≤4.
0
with weight less than or equal to t. By the previous theorem, Webeginwithcycliccodes.Considerall2-orbitsinZ ×Z .
5 9
9
They are that has dimension dim(C) = 35; however, as we have
mentioned, its minimum distance cannot be greater than 4.
Q(0,0), Q(0,1), Q(0,3), Q(1,0), Q(1,1), Q(1,2),
NowwefindanupperboundfordimensionswhenaPD-set
Q(1,3), Q(1,6).
includes the Frobenius automorphism. Consider the code C
8
We consider the 29-dimensional codes C1,...,C6 with with defining set
defining sets
D(C )=Q(0,0)∪Q(1,3)∪Q(1,7)∪Q(1,11).
8
D(C ) = Q(1,2)∪Q(1,6)
1
D(C2) = Q(1,1)∪Q(1,6) One may check that d(C8) = 6 and dim(C8) = 32, so it
is a 2-error correcting abelian code. In this case, Lemma 13
D(C ) = Q(1,2)∪Q(1,3)
3
does not may be applied; however, we shall see that Λ is a
D(C4) = Q(1,1)∪Q(1,3) PD-set, Take e ∈ A(3,15) an error vector with supp(e) =
D(C5) = Q(1,0)∪Q(1,2) {p1,p2} ⊆ Z3 ×Z15 and set pj = (p1j,p2j) for j = 1,2. By
D(C6) = Q(1,0)∪Q(1,1). analyzingthedistributionofthe2-orbitsintheplaneZ3×Z15
weseethat,first,onemayuseΛtotransformp (cid:55)→(0,0)and
1
Adirectcomputationshowsthatd(C )=5fori=1,...,6.
i p (cid:55)→ (x,y), with 12 ≤ y ≤ 14. Then (cid:104)T ,T (cid:105) finishes the
2 1 2
Since
task.
Γ(C ) ⊃ {(0,0),(0,1),(0,3),(1,0),(1,1),(1,2),
i
(1,3),(2,3)}
Now we deal with 3-PD-sets.
fori=1,...,6thenbyLemma13wehavethatΛisa2-PD- Lemma 15. Let C ≤ A(r ,r ) be an abelian t-error cor-
1 2
set for C with respect to Γ(C )c for i=1,...,6.
i i recting code with set of check positions Γ(C) given by (3). If
An easy inspection of the 2-orbits shows that C ,...,C
1 6 t ≥ 3, f[1] = r2, g[sn] = r1 and Γ(C) intersects all q-orbits
are the highest dimensional 2-permutation decodable codes. in Z ×Z , then Λ is a 3-PD-set with respect to Γ(C)c.
So, by using as information set Γ(C)c, we have achieved the r1 r2
upper bound for length 45 given in [13] with respect to the Proof:Takee∈A(r1,r2)anerrorvectorwithsupp(e)=
usual information set. {p1,p2,p3}⊆Zr1×Zr2 and set pj =(p1j,p2j) for j =1,2,3.
Now we shall construct two binary 2-error correcting non Then use (cid:104)T1,T2(cid:105) to transform p1 (cid:55)→ (a1,0),p2 (cid:55)→ (0,a2)
cyclic codes of length 45. The first one C7, having dimension and p3 (cid:55)→p, for some a1 ∈Zr1,a2 ∈Zr2 and p∈Zr1×Zr2.
dim(C ) = 31 will be permutation decodable with PD-set Now,byhypothesiswehavethatthereexistsanintegerisuch
7
(cid:104){T1, T2}(cid:105) with respect to the information set Γ(C7)c. The that qi·p ∈ Γ(C), then σi(a1,0) = (∗,0), σi(0,a2) = (0,∗)
second one, C , having dimension dim(C ) = 32 will be and σi(p) belong to Γ(C).
8 8
permutation decodable with PD-set Λ with respect to the
Example 16. We shall design binary 3-error-correcting cyclic
information set Γ(C )c.
8 codes with the highest dimension in length 65 such that Λ
In this case, we consider all 2-orbits in Z ×Z . They are
3 15 contains a 3-PD-set with respect to our information set. In
Q(0,0), Q(0,1), Q(0,3), Q(0,5), Q(0,7), this case, the 2-orbits in Z ×Z are
5 13
Q(1,0), Q(1,1), Q(1,2), Q(1,3), Q(1,5),
Q(0,0), Q(0,1), Q(1,0), Q(1,1), Q(1,2), Q(1,4), Q(1,7).
Q(1,6), Q(1,7), Q(1,10), Q(1,11).
One may check that the code C having defining set We consider the 40-dimensional codes C ,...,C with
7 1 4
defining sets
D(C ) = Q(0,3)∪Q(0,7)∪Q(1,0)∪Q(1,11)
7
D(C ) = Q(0,0)∪Q(0,1)∪Q(1,1)
has dimension dim(C )=31 and minimum distance d(C )= 1
7 7
6, so that, it is a 2-error correcting abelian code. The param- D(C ) = Q(0,0)∪Q(0,1)∪Q(1,2)
2
eters for Γ(C7) are f[1] = 8 > f[2] = 3, g[1] = 1 < g[2] = 3 D(C3) = Q(0,0)∪Q(0,1)∪Q(1,4)
and they yield
D(C ) = Q(0,0)∪Q(0,1)∪Q(1,7).
4
Γ(C ) = {(0,0),(0,1),(0,2),(0,3),(0,4),(0,5),
7
Adirectcomputationshowsthatd(C )=8fori=1,...,4.
i
(0,6),(0,7),(1,0),(2,0)}.
Since
Weclaimthat(cid:104)T , T (cid:105)isa2-PD-setwithrespecttoΓ(C )c.
1 2 7
Take e∈A(3,15) an error vector with supp(e)={p ,p }⊆ {(0,0),(0,1),(1,0),(1,1),(1,2),(3,2),(2,1)}⊂Γ(Ci)
1 2
Z ×Z and set p =(p1,p2) for j =1,2. Clearly, we may
3 15 j j j fori=1,...,4thenbyLemma15wehavethatΛisa3-PD-
use (cid:104)T1,T2(cid:105) to transform p1 (cid:55)→ (x1,0) and p2 (cid:55)→ (x2,y2), set for C with respect to Γ(C )c for i=1,...,4.
where x ,x ∈ Z and 0 ≤ y ≤ 7. Then, use (cid:104)T (cid:105) to i i
1 2 3 2 1 An easy inspection of the 2-orbits shows that C ,...,C
1 4
transform (x ,y )(cid:55)→(0,y ), and we are done.
2 2 2 are the highest dimensional 3-permutation decodable codes.
Note that the same argument may be applied to, for exam- So, by using as information set Γ(C)c, we have improved the
ple, the code
upper bound for length 65 (which is dimension 38) given in
D(C) = Q(0,3)∪Q(0,7)∪Q(1,0) [14, Table X] with respect to the usual information set.
10
ACKNOWLEDGEMENT
TheauthorssincerelythankprofessorA´ngeldelR´ıoforhis
helpful comments and suggestions.
REFERENCES
[1] S. D. Berman, SemisimplecyclicandAbeliancodes, Cybernetics, vol.
3,no.3,pp.21-30,1967
[2] J.J.Bernal,A´.delR´ıoandJ.J.Simo´n,Anintrinsicaldescriptionofgroup
codes,Des.CodesCrypto.,no.51,pp.289-300,2009.
[3] P. Camion, Abeliancodes, MRC Tech. Sum. Rep. no. 1059, Univ. of
Wisconsin,Madison,1970.
[4] H. Chabanne, Permutation decoding of abelian codes, IEEE Trans.
Inform.Theory,vol.38,no.6,pp.1826-1829,1992.
[5] W.C.Huffman,Codesandgroups,inV.S.Pless,W.C.HuffmanandR.
A.Brualdi(editors),HandbookofCodingTheory,vol.II.North-Holland,
Amsterdam,pp.1345-1440,1998.
[6] H.N.Ward,Quadraticresiduecodesanddivisibility,inV.S.Pless,W.
C. Huffman and R. A. Brualdi (editors), HandbookofCodingTheory
vol.I.North-Holland,Amsterdam,pp.827-870,1998,
[7] H. Imai, A theory of two-dimesional cyclic codes, Information and
Control,no.34,pp.1-21,1977.
[8] J.D.Key,T.P.McDonough,V.C.MavronPartialpermutationdecodingfor
codesfromfiniteplanes,EuropeanJournalofCombinatoricsno.26,pp.
665-682,2005.
[9] F. J. MacWilliams Permutation decoding of systematic codes, Bell
SystemTech.J.,no.43,pp.485-505,1964.
[10] F.J.MacWilliams,N.J.A.SloaneThetheoryoferror-correctingcodes,
North-Holland,Amsterdam,1983.
[11] V. S. Pless, W. C. Huffman and R. A. Brualdi (editors) Handbook of
CodingTheoryvol.I.North-Holland,Amsterdam,1998.
[12] S.Sakata,Ondeterminingtheindependentpointsetfordoublyperiodic
arraysandencondingtwo-dimensionalcycliccodesandtheirduals,IEEE
Trans.Inform.Theory,vol.IT-27,no.5,pp.556-565,1981.
[13] S.G.S.Shiva,K.C.FungandH.S.Y.Tan,Onpermutationdecodingof
binarycyclicdouble-error-correctingcodesofcertainlengths(Corresp.),
IEEETrans.Inform.Theory,vol.16,no.5,pp.641-643,1970.
[14] S.G.S.ShivaandK.C.Fung,Permutationdecodingoncertaintriple-
error-correctingbinarycodes,IEEETrans.Inform.Theory,vol.16,no.
5,pp.641-643,1970.