There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as \(E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle .\) We study a special construction of self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), and Doubly Regular Tournaments (DRT). We construct quasi self-dual codes over E, and Type IV codes, that is, quasi self-dual codes whose all codewords have even Hamming weight. All these codes can be represented as formally self-dual additive codes over \(\mathbb {F}_4.\) The classical invariant theory bound for the weight enumerators of this class of codes improves the known bound on the minimum distance of Type IV codes over E.

有一个4 阶局部环E，没有乘法标识，由生成器和关系定义为\(E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b ^2=b,\,ab=a,\, ba=b\rangle .\)我们研究了E上的自正交码的特殊构造，基于与两类关联方案相关的组合矩阵，强正则图（ SRG）和双重定期锦标赛（DRT）。我们在E和 Type IV 码上构造了拟自对偶码，即所有码字的汉明权重为偶数的拟自对偶码。所有这些代码都可以表示为\(\mathbb {F}_4.\)此类代码的权重枚举器的经典不变理论界限改进了类型 IV 代码在E上的最小距离的已知界限。