There is a special local ring $E$ of order $4,$ without identity for the multiplication, defined by $E=〈a,b|2a=2b=0,a^2=a,b^2=b,ab=a,ba=b〉.$ We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over $E,$ and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

中文翻译：

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非单位环上的 IV 型代码

有一个特殊的本地环$乙$有秩序的$4,$乘法没有恒等式，定义为$乙=〈\mathrm{一个},b|2\mathrm{一个}=2b=0,{\mathrm{一个}}^2=\mathrm{一个},b^2=b,\mathrm{一个}b=\mathrm{一个},b\mathrm{一个}=b〉.$我们研究了非交换局部环上线性码的代数结构，特别是它们的残差码和扭转码。我们引入了准自对偶码的概念$乙,$IV类码，即所有码字具有均匀汉明权重的准自对偶码。我们利用不变理论研究了这些编码的权重枚举数，并将它们分类为短长度。