~
This research was done 7iJhile the author 7iJaB visiting the Department of
Statistics, University of North Caro'Lina at Chapel Hitz, in May, 19'10, and 7iJaB
pa:rtiaU,y sponsored by the Army Research Office, Durhcun, under Grant No. DA-AROD-31-124-G910 and by the United States Air Foree Office of Scientific Research
under Contract No. AFOSR-68-1406.
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COMBINATORIAL
MATHEMATICS
YEA R
2
9
7
8
February 1969 - June 1970
ABELIAN CODES
,.
by
P. Carnian
~*
University of Toulouse
Department of Statistics
University of North Caro'Lina at Chapel Hill
Institute of Statistics Mimeo Series No. 600.32
July, 19'10
and M. R.
C~
University of Wisconsin.
Abelian Codes *
P. Camion
*If
Department of Statistics
University of North CaroLina at ChapeZ Hilt
B.C.H. Codes are ideals of the K-algebra
field and
when
G is a cyclic group.
K[G],
where
K is a finite
We call "Abelian Codes" the ideals of
G is any finite abelian group.
K[G]
For studying such ideals, we introduce a
tool which we have already used in [4] as "The Fundamental Isomorphism", in the
case where
G was cyclic.
referring to [6].
We call it here a Fourier Transform, briefly F.T.,
The definition of the F.T., in the general case, requires,
with a slight modification, the H. Mann's treatment of "Characters of Finite
Abelian Groups" [8] of which use in [7] is rather close to the present.
This leads to a generalization of B.C.H. theorem ot Abelian Codes, giving
their dimension and a lower bomd for their distances.
It is proved that all
Abelian Codes are neither isomorphic nor trivially obtained from Kronecker
products of cyclic codes.
Several classes of codes are given for which the exact values of distances
are established.
Finally, an example is worked out of binary code with length
49, dimension 18 and distance 12, which requires a special proof.
*
This research !Vas done whi Ze the author was visiting the Depar>tment of
University of North CaroLina at Chapel Hill~ in May~ 1970~ and was
partially sponsored by the Army Research Office~ Durham~ under Grant No. DA-AROD-31-124-G910 and by the United States Air Force Office of Scientific Research
under Contract No. AFOSR-68-1406.
** Univer8it7J of Toulouse and M.R. C., Unive!'Sity of Wisconsin.
Statistics~
2
The proofs which are omitted here will be found in the paper "Abelian
Codes" submitted to Information and Control..
1. CHARAcTERS OF /-\BalAN
~s.
(Based on H. Mann [3].)
1.1. The dual group of an abelian group.
Let
G be an additive group and let
multiplicative cyclic group of order
IGI
=v
into
[a}.
(l)
[a]
is the order of
G.
A
e,
[a]" {a, ••• ,ai, ••• ,ae"l}
where
cha:racter
X
of
We define the sum of two characters
(X+x')(g)"
x(g)X'(g),
Vg
e
be a
is the exponent of
G.
G is a homomorphism of
X and
G
X'
~ G.
being an abelian group, the sum of two characters is a character.
PROPOSITION
ditive group
1.
The set of cha:racters of a finite abelian group
G*~
isomorphic to G;
G is an ad-
G is itseZf the group of charaaters of
G*.
Outline of the argument.
G is the direct sum of cyclic groups
and if
gi
is a generator of
G ,
i
i " l, ••• ,s,
every
g
in
G may be
written uniquely as
where
ii is an integer modulo Vi
characters
(4)
j .. 1, •••
(Vi
III
IGil).
,s, by
i,j .. l, •.• ,s.
We then define
s
distinct
3
So every character may be written uniquely
where
hi
is an integer modulo
vi'
and the annolmced isomorphism is defined
by
(6)
8
\-..
i
xt'
i
III
l, •••
,s.
1.2. The monomial representation of G and G*.
From (2).
where
ii
G is isomorphic with the group of s-tuples with the form
runs over the set of integers modulo
an element of
G
viand we may thus represent
by the monomial
i
Xs
(2)
s '
To the sums of elements of
and so
G will correspond the product of monomials;
is represented by a group of monomials with the form
G*
h
Us
(briefly
s
Now let
r
i
na
(3)
III
e/v •
i
iihiri
If we substitute
=
to
UX) •
Xi
in (2),
we obtain
Ilihiri
ai
i
in (3).
and so do we by substituting
g
and
considered as a character of
X is the function (3).
G*
is thus the monomial function (2),
We shall now write
Xg or
gX
for
X(g) ... g(x).
4
2.
THE FouRIER TRANSFOR1 OF
2.1.
ft.sEBRAs OF
~alm
GRot.Fs OVER INTEGER I:bMAINS.
Integer Domain L.
Given an abelian group with exponent
e,
we consider any integral domain
L whose multiplicative monoid contains a cyclic subgroup
The oharacteristio of such a ring oou'td not divids
H of order e.
e.
We now consider the convolution L-algebra L[G].
2.2. The Fourier Transform.
Referring again to H. Mann [3], we
x ;:
(1)
L xgXg ,
an element of K[G]
g E: G.
g
for any subring K of
r~present
L.
So, every character is extended into a K-homomorphism of K[G]
into
L,
K[G]
by
by
(2)
X(x)
;: gL xgXg·
Now we define the Fourier Transform x
t
(briefly F.T.) of
x
E:
(3)
where
x
t
X
- X(x)
as defined by (2).
According to the structure involved,
second member of (3) or the e-tuple
X XE:
t
may be understood to be the
G*
G* E: L •
For evexry subring K of Ll~
into LG* •
PRoPOSITION
of KEG]
2.1.
t
(x)
x
the
F.T. is a K-isomorphism
2.3. The inversion formula.
The element
version formula
Lgxgxg
of
K[G)
may be obtained from its F.T. by the in-
5
1 \
t
-v L xX(Xg)
-1
•
X
Remark.
By 2.2 (3) and 1.2 (3).
vx
g
x.
g
may be obtained as the image of the
polynomial function 2.2 (3) for the argument corresponding to
-g.
2.4. Inject;vity of the Fourier Transform.
The following statements are easily proved:
2.1.
THEOREM
K[G]
into
If v- 1 belongs to L,
CoROllARY.
of L[G]
the F.T. is a K-isCPlozrphism of
For every sub:ring K of L"
LG* .
1l-
1
is the
F.T. and is an L-isomorphism
onto LG* •
2.4. The ideals of K[G], when L is a field, algebraic over K.
We have
THEOREM
2.3.
Every ideal of K[G]
is the dire~t sum of the minimal ideal that
it contains, considered as subspaces of K[G]
over K.
Every minimal ideal" considered as a subring of K[G]
some subfieZd of K(a).
is iSCPlozrphic to
The dimension over K of an ideal, T of K[G]
v - min I{xlx(x)
= oll
is
•
X€T
THEOREM 2.4.
A poz.yncmial. belongs to the F.T. of KeG]
if and only if it is
fixed by every Galois K-automorphism of L extended to L[G].
CoROLLARY.
The minimal supports of G*
g.. ·oup of the Euz.er group of e
Theorem 2.4 was proved for
are the orbits of G*
under the sub-
isomorphic to the GaZois group of Lover K.
G
cyclic in [4].
6
3.
3.1.
ABELl N4 ColES.
~Introduction.
Let
K
D
F
and let
2
G be the abelian group of type
from the corollary of Theorem 2.4 that
supports and hence that
K[ G]
has
2
G*
5
(3,3).
It follows
is partitioned into five minimal
ideals.
Now the number of distinct
ideals having a generator with the form
is
4
2 •
A code generated by a polynomial with the form
(l)~
we shall call a
separate cods.
3.2. Separate codes.
3.2.1.
The support polynomial.
To every ideal
C of
is the support of the ideal
poZynomiaZ of the codE
K[ G]
lJ.
-Ie.
corresponds a polynomial
LXE:J UX
where
We shall call such a polynomial the support
C as well as of its F.T.
ct.
We have
THEOREM
3.1.
The three foZleMing statements are equivalent.
1.
An abe lian code is separab Ze.
2.
An abelian code is the Kronecker product of cyclic codes.
3.
The support potynomiat of an abelian codE has the fo:rm
=
CoROUJ\RY.
Burton and WeZdon (1965) [3].
cods of 'length n
1
generated by b (1) •
generated by
a(X).
(n ,n 2 ) = 1. A is a cyclic
1
B is a cyclic code of length n 2
Let
Then the Kronecker product of A and B is a cyclic
code of length niI'L2 generated by
J
7
g(Z)
4. A GeNERAL STATEMENT
Let
feu)
...
a(Z
FOR THE
P2n2
p n
n n
)b(Z 1 l)mod(Z 1 2_ 1 ),
B.e.H.
THEOREM.
be the support polynomial of an abelian code.
d(X) denotes the degree of
dj (X)
cient of U
in f(X)(U)
UXf(U).
f(X)
in
U
and
j
Let
f(X)
f(X)(U) ...
is the coeffi-
j
j
The apparent distance
dfr(fJ
=
dic(f)
f
the set of all terms
rJ
and
0
d*(CJ
af'" f
of a code
f
feU)
is defined by
max
X€Gic
Now. given an abelian code
F
of a polynomial
C
with support polynomial
that may be obtained by writing
for every Galois automorphism.
C with support polynomial
u
J
will be
f€F
where
THEOREM
d~(f)
4.1.
is given by (2).
dfr(CJ
is a lobJer bound for the distance of
5. E'xAt-PLES OF ABELJPH CoDES.
5.1. Generalized first order Reed-Muller Codes.
The general support polynomial of such a code is
for
K ... Fq • L • FqV •
J
u
J
we denote by
.. f+g,
with
Then' the apparent distance
min dic(f)
(3)
u •
c.
8
The generalized first ord£r Reed-Mu'LZer codes~ idsa'Ls of F [GJ~
q
where G is the direct SW1l of s ~clic groups of order qV -1 have length
STATEfIENT
5.1.
(qV_1)8~
and dimension
into s+1
c'Lasses.
vs.
The
The
vectors of the code are partitioned
qVS
i-th class,
= o~ ... ~s~
i
contains
(~)(qV_l)i vectors of weight
(2)
The distance of the code is
qv-I (qv-2)(qv-1) s-2 (q-l).
(3)
CoROLLARY
length
5.1.
If
(v~q-1) = 1~
there exists a linear code over Fq with
dimension vs and 7M1:ghte and mstaneee gitten hy cb.'OP-
(q V -1) 8 I(q-l),
ping the factor- q-1 in
ExPJ.'flES.
For
with length
Here
q. 3,
n. 338,
k
v
and
(8)
co
3,
dimension
(3).
s = 2,
Corollary 5.1
k. 6
and distance
gives a code over
F3
d· 225.
reaches the Plotkin bound.
5.2. A binary non separable abelian code C of length 49. dimension 18
and distance 12.
The support polynomial of the code is
+
where
q. 2,
A last remark.
v· 3,
L
osl,h<v
s = 2.
(After d:tsCU8sion with H. Mann.)
H. Mann observes that an ideal of
K[G]
which is not a Kronecker product
of cyclic codes may become so by an automorphism of the group
G*
of characters
9
which would correspond to replacing each
U
i
by a monomial in the support
polynomial.
Now, for Example 5.2, it may be seen that it is certainly not possible to
exhibit such an automorphism of
G*.
REFERENCES
[1]
Albert, A., "Fundamental Concepts of Higher Albegra",
of Chicago Press, 1956.
[2]
Assmus, E.F., Jr. and H.F. Mattson, 1966, '~erfect Codes and the
Mathieu Groups", Arch. Math. 17, 121-135.
[3]
Berlekamp, E.R., "Algebraic Coding Theory", 1968, McGraw-Hill Book
Company, New York.
[4]
Carnion, P., 1968, "A Proof of Some Properties of R.M. Codes by Means
of the Normal Basis Theorem", Proceed. of the Conf. on Comb. Math.
(April 10-14, 1967), The Univ. of N.C. Press, Chapel Hill, N.C.
[5]
Carnion:l P., 1966, "Codes correcteurs d' erreur" ,
3 me Trimestre 1966 - Numero special.
[6]
Loomis, L.H., 1953,
Nostrand.
[7]
MacWilliams, F.J. and H.B. Mann, 1968,
Matrix of a Difference Set".
[8]
Mann, H.B., 1965,
[9]
Peterson, W.W..
"Abstract Harmonic Analysis",
"Addition Theorems",
The University
Revue du CETHEDEC
New York, Van
"On the p-Rank of the Design
Wiley & Sons.
"Error-Correcting Codes",
Wiley, 1962.