具有7-(8,0)型自同构的二元双偶极值码

Extremal self-dual doubly-even binary codes with type of 7-(8,0)

把长度为56的具有7-(8,0)型自同构的二元双偶极值码看成二元域上的多项式,分解成一个收缩码和一个偶重量多项式的直和。利用已知的码长较短的编码构造出长度为56的具有7-(8,0)型自同构的二元双偶极值码,并给出可能的生成矩阵。由于该双偶极值码的最小距离为12,根据两个码等价的类型,对生成矩阵运行Matlab程序,证明了在等价情况下,当E1(σ)*的维数K=2时,不存在有7-(8,0)型自同构的二元双偶极值码;当E1(σ)*的维数K=4时,有7-(8,0)型自同构二元双偶极值码共75种。对长度为56到70的具有类似型自同构的双偶极值码进行相似讨论,得到了其生成矩阵和分类情况。至此,长度在50到70之间的二元双偶极值码的生成矩阵和分类情况得到完全解决。

By regarding the extremal self-dual doubly-even binary codes with type of 7-（8,0） as a pol- ynomial over GF（2）＂ , we can factorize it to a contract code and even weight polynomials, and construct extremal self-dual doubly-even binary codes with type of 7-（8,0） by shorter codes. The possible genera- tor matrix of the codes is given. Matlab procedure implementation shows that, because the minimal dis tance is 12, codes with type of 7-（8,0） do not exist when the dimension of E1 （σ）＊ is 2 and there are 75 self-dual doubly-even binary codes with type of 7-（8,0） when the dimension of E1（σ）＊ is 4. We discuss the similarities of the extremal self-dual doubly-even binary codes with length of 56 to 70, and obtain the generator matrix and classification. So the generator matrices and classification of the extremal self-dual doubly-even binary codes with length between 50 and 70 can be solved completely.