One of the most important and challenging problems in coding theory is to construct codes with best possible parameters and properties. The class of quasi-cyclic (QC) codes is known to be fertile to produce such codes. Focusing on QC codes over the binary field, we have found 113 binary QC codes that are new among the class of QC codes using an implementation of a fast cyclic partitioning algorithm and the highly effective ASR algorithm. Moreover, these codes have the following additional properties: a) they have the same parameters as best known linear codes, and b) many of the have additional desired properties such as being reversible, LCD, self-orthogonal or dual-containing. Additionally, we present an algorithm for the generation of new codes from QC codes using ConstructionX, and introduce 35 new record breaking linear codes produced from this method.
翻译:在编码理论方面,最重要和最具挑战性的问题之一是建立具有最佳参数和属性的编码。已知,准周期(QC)编码类别具有生产这种编码的肥沃性。侧重于二进制域的QC编码,我们发现在QC编码类别中,113个二进制QC编码是新的,采用快速周期分割算法和高效的ASR算法。此外,这些编码还具有以下其他特性:(a) 它们具有与最著名的线性编码相同的参数,以及(b) 许多具有其他想要的特性,例如可逆性、LCD、自体或双层。此外,我们提出了一种算法,用于利用建筑X生成QC编码的新编码,并采用了从这种方法产生的35个新记录破线性编码。