Construction of self-dual binary [2²ⁿ,2²ⁿ⁻¹,2ⁿ]-codes
The binary Reed-Muller code RM(m−n,m) corresponds to the n-th power of the radical of GF(2)[G], where G is an elementary abelian group of order 2m. Self-dual RM-codes (i.e. some powers of the radical of the previously mentioned group algebra) exist only for odd m. The group algebra approach enables...
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| Cite this: | Construction of self-dual binary [2²ⁿ,2²ⁿ⁻¹,2ⁿ]-codes / C. Hannusch, P. Lakatos // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 59-68. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-1552032019-06-17T01:26:53Z Construction of self-dual binary [2²ⁿ,2²ⁿ⁻¹,2ⁿ]-codes Hannusch, C. Lakatos, P. The binary Reed-Muller code RM(m−n,m) corresponds to the n-th power of the radical of GF(2)[G], where G is an elementary abelian group of order 2m. Self-dual RM-codes (i.e. some powers of the radical of the previously mentioned group algebra) exist only for odd m. The group algebra approach enables us to find a self-dual code for even m=2n in the radical of the previously mentioned group algebra with similarly good parameters as the self-dual RM codes. 2016 Article Construction of self-dual binary [2²ⁿ,2²ⁿ⁻¹,2ⁿ]-codes / C. Hannusch, P. Lakatos // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 59-68. — Бібліогр.: 15 назв. — англ. 1726-3255 2010 MSC:94B05, 11T71, 20C05. http://dspace.nbuv.gov.ua/handle/123456789/155203 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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The binary Reed-Muller code RM(m−n,m) corresponds to the n-th power of the radical of GF(2)[G], where G is an elementary abelian group of order 2m. Self-dual RM-codes (i.e. some powers of the radical of the previously mentioned group algebra) exist only for odd m. The group algebra approach enables us to find a self-dual code for even m=2n in the radical of the previously mentioned group algebra with similarly good parameters as the self-dual RM codes. |
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Hannusch, C. Lakatos, P. Construction of self-dual binary [2²ⁿ,2²ⁿ⁻¹,2ⁿ]-codes Algebra and Discrete Mathematics |
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Hannusch, C. Lakatos, P. |
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Construction of self-dual binary [2²ⁿ,2²ⁿ⁻¹,2ⁿ]-codes |
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Construction of self-dual binary [2²ⁿ,2²ⁿ⁻¹,2ⁿ]-codes |
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Construction of self-dual binary [2²ⁿ,2²ⁿ⁻¹,2ⁿ]-codes |
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Construction of self-dual binary [2²ⁿ,2²ⁿ⁻¹,2ⁿ]-codes |
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Construction of self-dual binary [2²ⁿ,2²ⁿ⁻¹,2ⁿ]-codes |
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construction of self-dual binary [2²ⁿ,2²ⁿ⁻¹,2ⁿ]-codes |
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Construction of self-dual binary [2²ⁿ,2²ⁿ⁻¹,2ⁿ]-codes / C. Hannusch, P. Lakatos // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 1. — С. 59-68. — Бібліогр.: 15 назв. — англ. |
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Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 21 (2016). Number 1, pp. 59–68
© Journal “Algebra and Discrete Mathematics”
Construction of self-dual binary
[22k, 22k−1, 2k]-codes
Carolin Hannusch∗ and Piroska Lakatos
Communicated by V. I. Sushchansky
Abstract. The binary Reed-Muller code RM(m − k, m)
corresponds to the k-th power of the radical of GF (2)[G], where G
is an elementary abelian group of order 2m (see [2]). Self-dual RM-
codes (i.e. some powers of the radical of the previously mentioned
group algebra) exist only for odd m.
The group algebra approach enables us to find a self-dual code
for even m = 2k in the radical of the previously mentioned group
algebra with similarly good parameters as the self-dual RM codes.
In the group algebra
GF (2)[G] ∼= GF (2)[x1, x2, . . . , xm]/(x2
1
− 1, x2
2
− 1, . . . x2
m
− 1)
we construct self-dual binary C = [22k, 22k−1, 2k] codes with prop-
erty
RM(k − 1, 2k) ⊂ C ⊂ RM(k, 2k)
for an arbitrary integer k.
In some cases these codes can be obtained as the direct product
of two copies of RM(k −1, k)-codes. For k > 2 the codes constructed
are doubly even and for k = 2 we get two non-isomorphic [16, 8, 4]-
codes. If k > 2 we have some self-dual codes with good parameters
which have not been described yet.
∗Research of the first author was partially supported by funding of EU’s FP7/2007-
2013 grant No. 318202.
2010 MSC: 94B05, 11T71, 20C05.
Key words and phrases: Reed–Muller code, Generalized Reed–Muller code,
radical, self-dual code, group algebra, Jacobson radical.
60 Construction of self-dual binary [22k, 22k−1, 2k]-codes
Introduction and Notation
Let K be a finite field of characteristic p and let V be a vector space
over K, and C be a subspace of V. Then C is called a linear code. Let
x, y ∈ C, then the Hamming weight of x is the number of its non-zero
coordinates and the Hamming distance of x and y is the weight of x − y.
The Hamming distance (or weight) of a linear code C is the minimum of
all Hamming distances of its codewords.
In the study of binary codes C ⊆ V it is convenient that the space
V has an additional algebraic structure. For example, if V is a group
algebra K[G], where G is a finite abelian p-group and C is an ideal of
such a group algebra, then C is called an abelian group code.
The Hamming distance of a linear code determines the ability of
error-correcting property of the code. The authors in [6] proved that for
any 1 6 d 6
[
m+1
2
]
there exists an Abelian 2-group Gd that a power
of the radical is a self-dual code with parameters (2m, 2m−1, 2d). These
codes are ideals in the group algebra GF (2)[Gd] and they are “monomial
codes” in the sense of [5] as defined below.
Throughout, p will denote a prime and K a field of p elements. Let
G = 〈g1〉 × · · · × 〈gm〉 ∼= Cm
p be an elementary abelian p-group of order
pm i.e. K[G] is a modular group algebra, then the group algebra K[G]
and Kn are isomorphic as vector spaces by the mapping
ϕ : K[G] 7→ Kn, where ϕ
(
n
∑
i=1
aigi
)
7→ (a1, a2, . . . , an) := c ∈ C.
Reed-Muller (RM) binary codes were introduced in [12] as binary functions.
These codes are frequently used in applications and have good error
correcting properties. Now we are looking for self-dual codes in the radical
of K[G] with similarly good parameters as the RM codes.
If K is a field of characteristic 2 Berman [2] and in the general case
Charpin [3] proved that all Generalized Reed-Muller (GRM) codes coincide
with powers of the radical of the modular group algebra of K[G], where G
is an elementary abelian p-group. This group algebra is clearly isomorphic
with the quotient algebra
GF (p)[x1, x2, . . . xm]/(xp
1 − 1, . . . xp
m − 1).
Self-dual RM-codes (i.e. some power of the radical of the group algebra
GF (2)[G]) exist only for odd m. They are (2m, 2m−1, 2
m+1
2 )-codes.
C. Hannusch, P. Lakatos 61
For any basis {g1, g2, . . . gm} of such a group G consider the algebra
isomorphism µ mapping gj 7→ xj (1 6 j 6 m), and therefore we have the
algebra isomorphism
Ap,m
∼= GF (p)[x1, x2, . . . , xm]/(xp
1 − 1, xp
2 − 1, . . . xp
m − 1),
where GF (p)[x1, x2, . . . , xm] denotes the algebra of polynomials in m
variables with coefficients in GF (p).
It is known ([7]) that the set of monomial functions (ki ∈ N ∪ 0)
{
m
∏
i=1
(xi − 1)ki where 0 6 ki < p
}
form a linear basis of the radical Jp,m. Clearly the nilpotency index of
Jp,m (i.e. the smallest positive integer t, such that J t
p,m = 0) is equal to
t = m(p − 1) + 1.
Introducing the notation
Xi = xi − 1, (1 6 i 6 m)
(which will be used from now on) we have the following isomorphism
Jp,m ≃ GF (p)[X1, X2, . . . , Xm]/(Xp
1 , Xp
2 , . . . Xp
m). (1)
The k-th power of the radical consists of reduced m-variable (non-
constant) polynomials of degree at least k, where 0 6 k 6 t − 1, where
t = m(p − 1) + 1.
J k
p,m = GRM(t − 1 − k, m) = 〈
m
∏
i=1
(Xi)
ki |
m
∑
i=1
ki > k (0 6 ki < p)〉. (2)
Such a basis was exploited by Jennings [7].
By (2) the quotient space J k
p,m/J k+1
p,m has a basis
{
m
∏
i=1
Xki
i + J k+1
p,m , where 0 6 ki < p and
m
∑
i=1
ki = k
}
. (3)
Remark 1. It is known [15] that the dual code C⊥ of an ideal C in Ap,m
coincides with the annihilator of C∗, where C∗ is the image of C by the
involution ∗ defined on Ap,m by
∗ : g 7→ g−1 for all g ∈ G from Ap,m to itself.
The annihilator of J k
p,m is obviously J
m(p−1)+1−k
p,m . Thus the dual codes
of GRM-codes are GRM-codes and
GRM(k, m)⊥ = GRM(m(p − 1) − k − 1, m).
It follows that for m = 2k + 1 and p = 2 the code GRM(k, m) is self-dual.
62 Construction of self-dual binary [22k, 22k−1, 2k]-codes
1. Construction of binary self-dual codes
Let us consider the group algebra
A2,m = GF (2)[x1, . . . xm]/(x2
1 − 1, x2
2 − 1, . . . x2
m − 1) ≃ GF (2)[Cm
2 ]
as a vector space with basis
xa1
1 xa2
2 . . . xam
m , ai ∈ {0, 1}. (4)
It is known ([7]) that the radical J2,m of this group algebra is generated
by the monomials Xi = xi − 1 = xi + 1.
Definition 1 ([5]). The code C in J2,m (see (1)) is said to be a monomial
code if it is an ideal in A2,m generated by some monomials of the form
Xk1
1 Xk2
2 . . . Xkm
m , where 0 6 ki 6 1 (5)
The codes we intend to study are monomial codes.
For p = 2 using the usual polynomial product in the Boolean monomial
Xk1
1 Xk2
2 . . . Xkm
m (ki ∈ {0, 1}) we have
Xk1
1 Xk2
2 . . . Xkm
m = (x1 + 1)k1(x2 + 1)k2 . . . (xm + 1)km
and the Hamming weight in the basis (4) of this monomial equals
m
∏
i=1
(1+ki).
Example. Let G be an elementary abelian group of order 2m, m > 2.
Define the codes Cj as ideals in K[G] generated by Xj = xj − 1. These
codes are binary self-dual [2m, 2m−1, 2] codes and they are self-dual since
Cj = C⊥
j = 〈Xj〉. Further, this code is a direct sum of [2, 1, 2]-codes.
The dimension of the code Cj is 2m−1, the same as the dimension of the
radical of the group algebra GF (2)[H], where H is an elementary abelian
2-group of rank m − 1. The minimal distance of Cj is d = 2. This follows
from the fact that the element Xj = xj + 1 is included in the basis of Cj .
Thus, Cj is a self-dual [2m, 2m−1, 2]-code.
By Remark 1 one can see that a power of the radical of a modular
group algebra is self-dual if and only if the nilpotency index of the radical
is even. In our case (when G is elementary abelian of order pm) the
nilpotency index is even if and only if p = 2 and m is odd.
If m is odd, the binary RM-codes with parameters [2m, 2m−1, 2
m+1
2 ]
are self-dual and they are the m+1
2 -th powers of the radical A2,m.
C. Hannusch, P. Lakatos 63
For m = 2k where k is an arbitrary integer, we have a new method to
construct a doubly-even class of binary self-dual C codes with parameters
[2m, 2m−1, 2k]. For this code C we have RM(k − 1, 2k) ⊂ C ⊂ RM(k, 2k).
In the case of m = 4, we get two known extremal [16, 8, 4] codes (listed
in [14]) and for m > 4 these codes are not extremal. A doubly-even (i.e.
its minimum distance is divisible by 4) self-dual code is called extremal,
if we have for its minimum distance d = 4
[
n
24
]
+ 4, where n denotes the
code length (see Definition 39 and Lemma 40 in [8]).
To abbreviate the description of our codes, we shall refer to the mono-
mial Xk1
1 . . . Xkm
m as the m-tuple (k1, k2, . . . , km) ∈ {0, 1, . . . , p − 1}m of
exponents.
Using Plotkin’s construction of RM-codes (see Theorem 2 [13], Ch. 13,
§ 3) we obtain the following property of RM-codes.
Lemma 1. If m is even and m = 2k, then RM(k−1, m) = J k+1
2,m contains
a proper subspace which is isomorphic to RM(k − 1, m − 1).
Proof. Recall, that the set of monomials in the basis (2) of J k+1
2,m is
invariant under the permutations of the variables Xi, i.e. the set of binary
m-tuples (k1, k2, . . . , km) assigned to the basis (2) is invariant under the
permutation of all elements of the symmetric group Sm. Take the basis
elements with km = 1. Then the monomials Xk1
1 . . . Xkm
m of degree m can
be projected by π : (k1, k2, . . . , km−1, 1) 7→ (k1, k2, . . . , km−1). In this way
we get a basis of J k
2,m−1
∼= RM(k − 1, m − 1).
For m = 2k denote the set of all k-subsets of {1, 2, . . . , 2k} by X.
The elements of X can be described by binary sequences (k1, k2, . . . , km)
consisting of k ‘0‘-s and k ‘1‘-s in any order. Clearly, the cardinality of
the set X is
(2k
k
)
.
We say that the subset Y of binary m-tuples in X is complement free
if y ∈ Y implies 1 − y /∈ Y, where 1 = (1, 1, . . . , 1). Denote the set of
monomials corresponding to the set of exponents in X by X . Denote the
set with maximum number of pairwise orthogonal monomials in X by Y
and their corresponding exponents in X by Y.
Example. For m = 6 the quotient space J 3
2,m/J 4
2,m has a basis with
(6
3
)
= 20 elements, where the binary 6-tuples corresponding to the coset
64 Construction of self-dual binary [22k, 22k−1, 2k]-codes
representative monomials (the set X) are listed in pairs of complements:
(1, 1, 1, 0, 0, 0) (0, 0, 0, 1, 1, 1)
(1, 1, 0, 1, 0, 0) (0, 0, 1, 0, 1, 1)
(1, 1, 0, 0, 1, 0) (0, 0, 1, 1, 0, 1)
(1, 1, 0, 0, 0, 1) (0, 0, 1, 1, 1, 0)
(1, 0, 1, 1, 0, 0) (0, 1, 0, 0, 1, 1)
(1, 0, 1, 0, 1, 0) (0, 1, 0, 1, 0, 1)
(1, 0, 1, 0, 0, 1) (0, 1, 0, 1, 1, 0)
(1, 0, 0, 1, 1, 0) (0, 1, 1, 0, 0, 1)
(1, 0, 0, 1, 0, 1) (0, 1, 1, 0, 1, 0)
(1, 0, 0, 0, 1, 1) (0, 1, 1, 1, 0, 0)
and we have 2
1
2 (6
3) = 210 complement-free sets. For example the following
complement free sets Y and Y of 10 elements:
Y Y
(1, 1, 1, 0, 0, 0), X1X2X3
(0, 0, 1, 0, 1, 1), X3X5X6
(1, 1, 0, 0, 1, 0), X1X2X5
(0, 0, 1, 1, 1, 0), X3X4X5
(1, 0, 1, 1, 0, 0), X1X3X4
(0, 1, 0, 1, 0, 1), X2X4X6
(0, 1, 0, 1, 1, 0), X2X4X5
(0, 1, 1, 0, 0, 1), X2X3X6
(1, 0, 0, 1, 0, 1), X1X4X6
(1, 0, 0, 0, 1, 1), X1X5X6
Theorem 1. Let C be a binary code with RM(k−1, 2k) ⊂ C ⊂ RM(k, 2k)
with the following basis of the factorspace C/RM(k − 1, 2k)
{
m
∏
i=1
Xki
i + RM(k − 1, 2k), where ki ∈ {0, 1} and
m
∑
i=1
ki = k
}
, (6)
where the set of the exponents (k1, k2, . . . , km) is a maximal (with cardinal-
ity 2
1
2 (2k
k
) ) complement free subset of X. Then C forms a [22k, 22k−1, 2k]
self-dual doubly-even code.
Proof. Let G be an elementary abelian group of order 2m, where m =
2k, k > 2. By the group algebra representation of RM-codes and the
definition of C we have the relation J k+1
2,m ⊂ C ⊂ J k
2,m.
C. Hannusch, P. Lakatos 65
For m = 2k the set X is the set of coset representatives of the quotient
space J k
2,m/J k+1
2,m , i.e. the set of monomials satisfying (6).
Clearly, two monomials Xk1
1 Xk2
2 . . . Xkm
m and X l1
1 X l2
2 . . . X lm
m are or-
thogonal, i.e. their product is zero, if for some i : 1 6 i 6 m we have ki = li.
Thus, the elements in the radical corresponding to these monomials
are orthogonal if their exponent m-tuples belong to a complement free
set.
The m-tuples (k1, k2 . . . km) have to be complement free in Y, otherwise
the corresponding monomials in Y are not orthogonal. Clearly Y is a
complement free subset of X (given by (4)) with cardinality 1
2
(2k
k
)
=
(2k−1
k−1
)
.
By definition, C = 〈 J k+1
2,m
⋃
Y 〉 is a subspace of the radical J2,m of
the group algebra A2,m generated by the union of J k+1
2,m and Y. For the
dimension of C we have
dim(C) = dim(RM(k−1, m))+
1
2
(
2k
k
)
= 1+
k−1
∑
i=1
(
2k
i
)
+
1
2
(
2k
k
)
= 22k−1.
It follows that C is self-dual. Since a binary self-dual code contains a word
of weight 2 if and only if the generator matrix has two equal columns, we
have our self-dual code to be doubly-even.
Each monomial in Y has the same weight 2k, that is the minimal
distance of C. Using the identities for the monomials involved in the basis
of our codes
xi(xj + 1) = (xi + 1)(xj + 1) + (xj + 1) and (xi + 1)2 = 0,
we easily obtain that C (which is subspace of J2,m) is an ideal in the
group algebra GF (2)[G].
Theorem 2. Let Y and Y be sets defined above and let C be the code
defined in Theorem 1. Suppose that ki = 0 for some i : 1 6 i 6 m in each
element of the subset Y, (i.e. the variable Xi is missing in each monomial
of Y). Then we have the isomorphism
C ≃ RM(k − 1, 2k − 1) ⊕ RM(k − 1, 2k − 1).
Proof. The elements of Y are of the form
Xk1
1 . . . Xkm
m = (x1 + 1)k1(x2 + 1)k2 . . . (xm + 1)km , where
m
∑
i=1
ki = k
66 Construction of self-dual binary [22k, 22k−1, 2k]-codes
and their weight is 2k. Project the set of monomials with ki = 0 in
C = 〈 J k+1
2,m
⋃
Y 〉 onto the monomials Xk1
1 , . . . , X
ki−1
i−1 , X
ki+1
i+1 , . . . , Xkm
m .
The image C1 of this projection is a self-dual RM(k − 1, 2k − 1)-code with
parameters [22k−1, 22k−2, 2k].
By Lemma 1 the elements of the basis of Jk+1
2,m with ki = 1 generate a
subspace C2 which is isomorphic to RM(k − 1, 2k − 1). The intersection of
C1 and C2 is empty. Therefore C ≃ C1⊕C2 and the statement follows.
Remark 2. In particular, by Theorem 1 we get [16, 8, 4] self-dual codes
for m = 4. These codes are extremal doubly-even codes. Using the SAGE
computer algebra software we may check easily the classification of binary
self-dual codes listed in [14].
There are two cases:
1) If ki = 0 for some i : 1 6 i 6 m in each element of the set Y , then
we get the direct sum E8 ⊕ E8, where E8 is the extended Hamming
code.
2) otherwise we get an indecomposable [16, 8, 4] code (which is denoted
by E16 in [14]).
These codes are formally self-dual. Both classes have the following
weight function:
z16 + 28z12 + 198z8 + 28z4 + 1
Remark 3. It is known that for each odd m > 1 there exists a self-dual
affine-invariant code of length 2m over GF (2), which is not a self-dual
RM-code [4].
The factor space J k
p,m/J k+1
p,m is an irreducible AGL(m, GF (p)) module.
Thus the code C is not affine invariant (see [1] Theorem 4.17 ) as the
powers of the radical of Ap,m are. The code C cannot be an extended
cyclic code by Corollary 1 in [4].
Remark 4. Using the inclusion-exclusion principle a formula can be
given for the dimension of the RM(k + 1, m)-code (see for example in [1]
Theorem 5.5). If p = 2 and 0 6 k 6 m, then we have
dim C = 1
2
(2k
k
)
+
m
∑
i=k+1
2k
∑
j=0
(−1)j
(2k
j
)(2k−2j+i−1
i−2j
)
=
m
∑
i=k+1
(2k
i
)
+ 1
2
(2k
k
)
,
where i − 2j > 0.
The codes constructed in the current paper are worth to be studied
further. Already for k = 2 we get two non-isomorphic codes with the
same parameters. It would be interesting to determine all classes of codes
C. Hannusch, P. Lakatos 67
up to isomorphism for each arbitrary integer k and to determine their
automorphism group. The code C in Theorem 1 is not affine-invariant
and first computations show that the automorphism group of C with
ki = 0 differs from the automorphism group of C with ki = 1 for some
1 6 i 6 m.
We can formulate the following open questions about the code C of
Theorem 1:
1) Does there exist a classification for all complement-free sets for
arbitrary even m?
2) How many non-equivalent (in any sense) self-dual binary codes exist
for fixed m and p?
3) Compare the automorphism groups of the codes C defined in Theo-
rem 1 with the automorphism group of RM-codes.
4) Find decoding algorithms for C.
References
[1] Assmus, E.F. Key, J.K., Polynomial codes and finite geometries, Chapter in Hand-
book of Coding Theory, edited by V. Pless and W. C. Huffman, Elsevier, 1995.
[2] Berman, S.D., On the theory of group code, Kibernetika, 3(1) (1967), 31–39.
[3] Charpin, P., Codes cycliques étendus et idA©aux principaux d’une alge‘bre modu-
laire, C.R. Acad. Sci. Paris, 295(1) (1982), 313–315.
[4] Charpin, P, Levy-Dit-Vehel, F., On Self-Dual Affine-lnvariant Codes Journal
Combiunatorial Theory, Series A 67 (1994), 223–244.
[5] Drensky, V., Lakatos, P., Monomial ideals, group algebras and error correcting
codes, Lecture Notes in Computer Science, Springer Verlag, 357 (1989), 181–188.
[6] Hannusch, C., Lakatos, P., Construction of self-dual radical 2-codes of given
distance, Discrete Math., Algorithms and Applications, 4(4) (2012).
[7] Jennings, S. A., The structure of the group ring of a p-group over modular fields,
Trans. Amer. Math. Soc. 50 (1941), 175–185.
[8] Joyner, D., Kim, J.-L., Selected unsolved problems in Coding Theory, Birkha̋user,
2011.
[9] Kasami, T. , Lin, S, Peterson, W.W., New generalisations of the Reed-Muller codes,
IEEE Trans. Inform. Theory II-14 (1968) 189–199.
[10] Kelarev, A. V.; Yearwood, J. L.; Vamplew, P. W., A polynomial ring construction
for the classification of data, Bull. Aust. Math. Soc. 79 , 2 (2009) 213–225.
[11] Landrock, P., Manz, O., Classical codes as ideals in group algebras, Designs, Codes
and Cryptography, 2(3) (1992), 273–285.
[12] Muller, D. E., Application of boolean algebra to switching circuit design and to
error detection, IRE Transactions on Electronic Computers, 3:6–12 (1954).
[13] MacWilliams, F.J., Sloane, N.J.A., The Theory of Error-Correcting Codes, North
Holland, Amsterdam, 1983.
68 Construction of self-dual binary [22k, 22k−1, 2k]-codes
[14] Pless, V., A classification of self-orthogonal codes over GF(2), Discrete Mathematics
3 (1972), 209–246.
[15] MacWilliams, F.J., Codes and Ideals in group algebras, Univ. of North Carolina
Press, 1969.
Contact information
C. Hannusch,
P. Lakatos
Institute of Mathematics, University of Debrecen, 4010
Debrecen, pf.12, Hungary
E-Mail(s): carolin.hannusch@science.unideb.hu,
lakatosp@science.unideb.hu
Web-page(s): www.mat.unideb.hu
Received by the editors: 21.09.2015
and in final form 16.12.2015.
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