There is a local ring $E$ of order $4,$ without identity for the multiplication, defined by generators and relations as $$E=〈a,b|2a=2b=0,a^2=a,b^2=b,ab=a,ba=b〉.$$ We study a recursive construction of self-orthogonal codes over $E.$ We classify, up to permutation equivalence, self-orthogonal codes of length $n$ and size $2^n$ (called here quasi self-dual codes or QSD) up to the length $n=12$. In particular, we classify Type IV codes (QSD codes with even weights) up to $n=12$.

中文翻译：

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非单位环上拟自对偶码的构建

有本地环$乙$有秩序的$4,$乘法没有恒等式，由生成器和关系定义为$$乙=〈\mathrm{一个},b|2\mathrm{一个}=2b=0,{\mathrm{一个}}^2=\mathrm{一个},b^2=b,\mathrm{一个}b=\mathrm{一个},b\mathrm{一个}=b〉.$$我们研究了自正交码的递归构造$乙.$我们对长度的自正交码进行分类，直至置换等价$n$和大小$2^n$（这里称为准自对偶码或 QSD）长达$n=12$. 特别是，我们将类型 IV 码（具有偶数权重的 QSD 码）分类为$n=12$.