There is a local ring I of order 4, without identity for the multiplication, defined by generators and relations as

We give a natural map between linear codes over I and additive codes over \({\mathbb{F}}_{4},\) that allows for efficient computations. We study the algebraic structure of linear codes over this non-unital local ring, their generator and parity-check matrices. A canonical form for these matrices is given in the case of so-called nice codes. By analogy with \({\mathbb{Z}}_{4}\)-codes, we define residue and torsion codes attached to a linear I-code. We introduce the notion of quasi self-dual codes (QSD) over I, and Type IV I-codes, that is, QSD codes all codewords of which have even Hamming weight. This is the natural analogue of Type IV codes over the field \({\mathbb{F}}_{4}.\) Further, we define quasi Type IV codes over I as those QSD codes with an even torsion code. We give a mass formula for QSD codes, and another for quasi Type IV codes, and classify both types of codes, up to coordinate permutation equivalence, in short lengths.

有一个4阶的局部环I，没有乘法的标识，由生成器和关系定义为

我们给出了I上的线性代码和\（{\ mathbb {F}} _ {4}，\）上的加性代码之间的自然映射，从而可以进行有效的计算。我们研究了在该非单位局部环上的线性代码的代数结构，它们的生成器和奇偶校验矩阵。在所谓的尼斯代码的情况下，给出了这些矩阵的规范形式。类似于\（{\ mathbb {Z}} _ {4} \）-代码，我们定义了附加在线性I代码上的残差和扭转代码。我们介绍I上的准自对偶码（QSD）和IV型I码的概念，即，所有码字均具有汉明权重的QSD码。这是现场IV型代码的自然类似物\（{\ mathbb {F}} _ {4}。\）。此外，我们将I上的准IV型代码定义为具有均匀扭转代码的QSD代码。我们为QSD码提供了一个质量公式，为准IV型编码提供了一个质量公式，并对这两种类型的编码进行了分类，以协调较短的长度。