16 research outputs found
Construction of quasi-cyclic self-dual codes
There is a one-to-one correspondence between -quasi-cyclic codes over a
finite field and linear codes over a ring . Using this correspondence, we prove that every
-quasi-cyclic self-dual code of length over a finite field
can be obtained by the {\it building-up} construction, provided
that char or , is a prime , and
is a primitive element of . We determine possible weight
enumerators of a binary -quasi-cyclic self-dual code of length
(with a prime) in terms of divisibility by . We improve the result of
[3] by constructing new binary cubic (i.e., -quasi-cyclic codes of length
) optimal self-dual codes of lengths (Type I), 54 and
66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and
60. When , we obtain a new 8-quasi-cyclic self-dual code
over and a new 6-quasi-cyclic self-dual code over
. When , we find a new 4-quasi-cyclic self-dual
code over and a new 6-quasi-cyclic self-dual code
over .Comment: 25 pages, 2 tables; Finite Fields and Their Applications, 201
Additive Self-Dual Codes over GF(4) with Minimal Shadow
We define additive self-dual codes over
G
F
(
4
)
with minimal shadow, and we prove the nonexistence of extremal Type I additive self-dual codes over
G
F
(
4
)
with minimal shadow for some parameters
MDS Self-Dual Codes and Antiorthogonal Matrices over Galois Rings
In this study, we explore maximum distance separable (MDS) self-dual codes over Galois rings G R ( p m , r ) with p ≡ − 1 ( mod 4 ) and odd r. Using the building-up construction, we construct MDS self-dual codes of length four and eight over G R ( p m , 3 ) with ( p = 3 and m = 2 , 3 , 4 , 5 , 6 ), ( p = 7 and m = 2 , 3 ), ( p = 11 and m = 2 ), ( p = 19 and m = 2 ), ( p = 23 and m = 2 ), and ( p = 31 and m = 2 ). In the building-up construction, it is important to determine the existence of a square matrix U such that U U T = − I , which is called an antiorthogonal matrix. We prove that there is no 2 × 2 antiorthogonal matrix over G R ( 2 m , r ) with m ≥ 2 and odd r
Additive Self-Dual Codes over GF(4) with Minimal Shadow
We define additive self-dual codes over
G
F
(
4
)
with minimal shadow, and we prove the nonexistence of extremal Type I additive self-dual codes over
G
F
(
4
)
with minimal shadow for some parameters
On the Problem of the Existence of a Square Matrix U Such That UUT = βI over Zpm
Building-up construction is one of several methods for constructing self-dual codes. Recently, a new building-up construction method has been developed by S. Han, in which the existence of a square matrix U such that U U T = - I is essential. In this paper, we completely solve the existence problem for U over Z p m , where p is an arbitrary prime number
On the Problem of the Existence of a Square Matrix U Such That UUT = βI over Zpm
Building-up construction is one of several methods for constructing self-dual codes. Recently, a new building-up construction method has been developed by S. Han, in which the existence of a square matrix U such that U U T = - I is essential. In this paper, we completely solve the existence problem for U over Z p m , where p is an arbitrary prime number
Near-Extremal Type I Self-Dual Codes with Minimal Shadow over GF(2) and GF(4)
Binary self-dual codes and additive self-dual codes over GF(4) contain common points. Both have Type I codes and Type II codes, as well as shadow codes. In this paper, we provide a comprehensive description of extremal and near-extremal Type I codes over GF(2) and GF(4) with minimal shadow. In particular, we prove that there is no near-extremal Type I [24m,12m,2m+2] binary self-dual code with minimal shadow if m≥323, and we prove that there is no near-extremal Type I (6m+1,26m+1,2m+1) additive self-dual code over GF(4) with minimal shadow if m≥22