Розширені бінарні коди Голея за груповою алгеброю однієї групи

Abstract

Extended binary Golay codes are examples of extreme binary self-dual codes of Type II (linear binary self-dual codes with Hamming distance between arbitrary codewords which are multiples of 4 that has the highest possible minimum Hamming distance among such codes with a fixed dimension of codeword space and their length). These codes have been studied for a long time and many different constructions have been established to build these codes. In addition, extended binary Golay codes are easy to obtain from binary Golay codes and vice versa. The latter are perfect codes and together with binary codes they give us all possible parameters of nontrivial binary perfect codes.In the paper the construction of linear binary codes, in particular of binary Golay codes extended by the group algebra $\mathbb{F}_2G$ of finite group $G=(C_6 \times C_2)~\rtimes~C_2$ of order $n=24$ over the field of two elements $\mathbb{F}_2$ has been considered. Extended binary Golay code is defined as any binary linear code, for which the length of the codewords is 24, the dimension of the subspase of the codewords is 12 and the minimum Hamming distance of the code is 8, that is, any [24,12,8]-code. Considering these codes, we apply the elements of the presentation theory, in particular regular presentations $v\to \sigma (v)$ of algebra $\mathbb{F}_2G$. For the element $v$ we define the binary code $C(v)$ as the subspace of $\mathbb{F}_2^{n}$ generated by the rows of the matrix $\sigma (v)$. The criterion of self-dual codes $C(v)$ for an arbitrary finite group $G$ of order 24 was used and easily verified necessary conditions for binary code $C(v)$ for elements $v$ of group algebra $\mathbb{F}_2G$ of the group $G=(C_6 \times C_2)~\rtimes~C_2$ to be self-dual was found. As a result of numerical calculations which involves verifying the found necessary conditions, we get the number of elements $v\in\mathbb{F}_2G$ that $C(v)$ is self-dual. Quantitative results for comparison with the number of the same elements when $v=v^*$ are presented. Previously, in this form, extended binary Golay codes were found only for elements $v$ that $v=v^*$. As a result of calculations we obtained all 27~648 elements $v$ of group algebra $\mathbb{F}_2G$ that $C(v)$ is extended binary Golay code.