Linear code on a finite field is a subspace of numerical vector space on a finite field. A quasi-cyclic code, the subject of this paper, is one in which the code word is divided into component segments of fixed length, each of which is invariant under cyclic permutation. One of the indicators to measure the error correction capability of a code is the minimum distance. A code whose minimum distance achieves an upper bound is called an optimal code, and a code that is close to optimal is called a good code. It is known empirically that quasi-cyclic codes give good codes, and for this reason many studies have been conducted to elucidate quasi-cyclic codes.

In this paper, the conditions for a given quasi-cyclic code to have reversibility, self-duality, and self-orthogonality are given in a simple formula using the generator polynomial matrix of the code. In addition, using these conditions, the optimal codes are constructed by computer search and a list is presented.

The representation method of quasi-cyclic code using generator polynomial matrices is an original feature of this paper.

The method presented here is useful for providing necessary and sufficient conditions for whether or not a quasi-cyclic code has a certain property.

The authors show that the representation method using generator polynomial matrices is a natural way to describe quasi-cyclic codes, and it can be said that this paper presents a new way to study quasi-cyclic codes.

It is also empirically known that quasi-cyclic codes give good codes, as mentioned earlier, and it is expected that this mechanism will be clarified through the method described in this paper. In addition, reversible codes can be applied to error correction for DNA storage, and future developments are expected.

In addition, since these equations are in a form that can be easily processed using a computer, they can be easily used as determination conditions in computer searches. They are also low-cost in terms of computational cost, making them very useful as a determination method. In addition, the authors showed that when the number of segments of a quasi-cyclic code is odd, it cannot be reversible, self-dual quasi-cyclic code, which has significant theoretical utility.