ISIT 2004, Chicago, USA, June 27 – July 2, 2004
Vandermonde-Cocyclic Codes and a suitable DFT
Kiran .T and B. Sundar Rajan
Dept. ECE, Indian Institute of Science, India
{kirant@protocol., bsrajan@}ece.iisc.ernet.in
Abstract — A new transform is defined for the class
of cocyclic group ring codes over a Galois ring, where
the underlying group is Abelian and the cocycle is
the Vandermonde-cocycle on the Abelian group. The
class of Vandermonde-cocyclic (VC) group ring codes
are characterized using this new transform by means
of a structure theorem for VC group rings.
For a prime p, let GR(pa, l) denote the Galois ring with
∗
denote the unique cyclic subcharacteristic pa [1]. Let Ta,l
group of order pl − 1 in the group of units GR(pa , l)∗ . Let
G, a finite Abelian group (with |G| = n), be a direct product of r cyclic subgroups denoted by Cr−1 , . . . , C0 of orders mr−1 , . . . , m0 respectively. Any element g ∈ G can
be denoted as gi or gi , where i = ir−1 , ir−2 , · · · , i0 is the mixed-radix representation of i ∈ In = {0, . . . , n −
1} using mr−1 , mr−2 , . . . , m0 as the mixed-radixes and the
group operation of G can also be specified using the mixedradix addition [2].
We consider linear codes of length
n = mr−1 . . . m1 m0 over the Galois ring GR(pa, l) such that
gcd(p, n) = 1. We use q for pl for notational simplicity.
∗
is called a cocycle on G
A function ψ : G × G → Ta,l
if ψ(x, y)ψ(xy, z) = ψ(y, z)ψ(x, yz) for all x, y, z ∈ G [3].
Let β̄ = (βr−1 , βr−2 , . . . , β0 ), where each βλ is an ele∗
, such that dλ (the order of βλ ) divides mλ
ment of Ta,l
for all λ = 0, 1,Q
. . . , r − 1. The map on G × G defined as
iλ jλ
, for i = ir−1 , ir−2 , . . . , i0 and
ψβ̄ (gi , gj ) = r−1
λ=0 βλ
j = jr−1 , jr−2 , . . . , j0 , is a cocycle. We call this cocycle as
Vandermonde-cocycle on an Abelian group. For any cocycle ψ
on G, let GR(pa, l)ψ [G] be the twisted group ring which has
the same module structure as the group ring GR(pa, l)[G], but
the multiplication is twisted, which is a linear extension of
g.h = ψ(g, h)gh for all g, h ∈ G. A cocyclic group ring code is
an ideal in GR(pa, l)ψ [G]. Vandermonde-cocyclic (VC) group
ring codes are ideals in the cocyclic group ring GRψβ̄ (pa, l)[G].
Let m be the smallest positive integer such that mλ | (q m − 1)
and also 2dλ | (q m − 1) for all λ = 0, 1, . . . , r − 1. Let
αλ = ζ sλ be a primitive root of unity of order mλ and
γλ = ζ hλ be a square-root of βλ , for all λ, belonging to
∗
= ζ ⊂ GR(pa, lm).
Ta,lm
Definition 1 (VC-DFT) Let a = (a0 , a1 , . . . , an−1 ) ∈
GR(pa, l)n and let αλ and γλ be as defined above. The VC = (A0 , A1 , . . . , An−1 ) ∈ GR(pa, lm)n of a is
DFT vector A
Pn−1 “Qr−1 −i2λ iλ jλ ”
defined as Aj =
ai for all j ∈ In ,
αλ
i=0
λ=0 γλ
where i = ir−1 , ir−2 , · · · , i0 and j = jr−1 , jr−2 , · · · , j0 are
mixed-radix representations of i and j.
λ=0
aik bk ψβ̄ (gik , gk )
k=0
r−1
Y
vc−df t
←→
Cj = Bj Aj ,
!
αiλλ
ai
vc−df t
←→
Aj⊕1,1,...,1 ,
λ=0
ci = ai bi
where
(u)
Ak
=
Pn−1 “Qr−1
i=0
iλ kλ
λ=0 αλ
vc−df t
←→
n−1
1X
(u)
Bk Ajk
n
k=0
”
ai is the Generalized DFT
for Abelian codes [2] and ⊕, denote the mixed-radix addition
and subtraction respectively.
Let σ0 denote the Frobenius automorphism of GR(pa , lm)
∈ GR(pa, lm)n is the VCand let σ = σ0l . A vector A
a
n
DFT vector of a ∈ GR(p , l) , if and only if σ k (Aj ) = Ai
where
i = ir−1 , i”r−2 , . . . , i0 is such that, iλ = φkλ (jλ ) =
“
−(q k −1)hλ
sλ
+ q k jλ
modulo mλ for all λ. This is the con-
jugate symmetry property. Let Φk denote the mapping
from In to In , which maps i = ir−1 , . . . , i0 to Φk (i) =
k
k
c =
For¯ every j ∈ In , the set j
φ
˘ r−11(ir−1 ),2 . . . , φ0 (i0 ).
j, Φ (j), Φ (j), . . . , Φej −1 (j) where ej is the smallest integer
such that Φej (j) = j, is called the cyclotomic coset containing
j and, ej is called the exponent of j. Let L ⊆ In be the
set containing one element from each of the cyclotomic cosets.
c
bi = {Aj | j ∈ i}
The set of transform components A
will be called the conjugacy class containing Ai . For a code
C over GR(pa, l), let Cj = {Aj | ∀a ∈ C} denote the set of
distinct values taken by the j-th transform component of all
the codewords in C and let Ci,j = {(Ai , Aj ) | ∀a ∈ C}.
Theorem 1 (Structure theorem) For any integer n =
gcd(p, n) = 1, GR(pa , l)n is isomormr−1 mr−2
L . . . m0 and
a
group ring code
phic to
i∈L GR(p , lei ). Further, a β̄-VC L
C over GR(pa, l) is isomorphic to an ideal
i∈L Ci , where
Ci = pηi GR(pa, lei ) for some fixed value of ηi , 0 ≤ ηi ≤ a and
transform components belonging to different conjugacy classes
take values independently.
Here, Ai and Aj take values independently implies Ci,j =
Ci × C j .
Transform Domain Characterization: A code over Galois
ring GR(pa , l) is β̄-VC iff the VC-DFT vector of each codeword
satisfies the conjugate symmetry property and for any i ∈ In ,
Ci = pηi GR(pa, lei ) for some fixed value of ηi , 0 ≤ ηi ≤ a.
References
[2] Kiran.T and B. Sundar Rajan, “Abelian codes over Galois rings
closed under certain permutations,” IEEE Trans. Inform. Theory, vol. 49, no. 9, Sept 2003.
λ=0
work was partly funded by the DRDO-IISc Program on
Advanced Research in Mathematical Engineering through a grant
to B.S.Rajan.
‹,(((
n−1
X
[1] McDonald B.R, Finite Rings with Identity, Marcel Dekker,Inc.,
New York, 1974.
Lemma 1 (Properties of VC-DFT)
!
!
r−1
r−1
Y i
Y
vc−df t
jλ
λ
βλ ai1,1,...,1
←→
γλ αλ Aj ,
1 This
ci =
[3] K.J. Horadam and A.A.I. Perera, “Codes from cocycles,” in
AAECC-12, T. Mora and H. Mattson, Eds., 1997, LNCS 1255,
pp. 151–163.