These are notes on a preliminary follow-up to a question of Jim Propp, about cyclic sieving of cyclic codes. We show that two of the Mahonian polynomials are cyclic sieving polynomials for certain Dual Hamming Codes: X and X inv for q = 2, 3 and q = 2, respectively.

It is well known that the problem of determining the weight distributions of families of cyclic codes is, in general, notoriously difficult. An even harder problem is to find characterizations of families of cyclic codes in terms of their weight distributions. On the other hand, it is also well known that cyclic codes with few weights have a great practical importance in coding theory and crypt...

Using Gröbner basis techniques, we can exhibit a method to get distance and weight distribution of both cyclic codes and shortened cyclic codes, extending and improving previous work on the distance of cyclic codes.

We completely characterize possible indices of quasi-cyclic subcodes in a cyclic code for a very broad class of cyclic codes. We present enumeration results for quasi-cyclic subcodes of a fixed index and show that the problem of enumeration is equivalent to enumeration of certain vector subspaces in finite fields. In particular, we present enumeration results for quasi-cyclic subcodes of the si...

We propose a new method to construct quantum synchronizable codes from classical cyclic codes using the idea of augmented cyclic codes. The method augments a dual-containing cyclic code C2 to obtain another cyclic codes C1 of higher dimension. The resulting two cyclic codes and the dual code C⊥ 2 satisfy the containing property C⊥ 2 ⊂ C2 ⊂ C1. The proposed construction method based on general q...

In this paper we study a special type of linear codes, called skew cyclic codes, in the most general case. This set of codes is a generalization of cyclic codes but constructed using a non-commutative ring called the skew polynomial ring. In previous works these codes have been studied with certain restrictions on their length. This work examines their structure for an arbitrary length without ...

Abstract In this paper some good quasi-cyclic codes over GF(3) are presented. These quasi-cyclic codes improve the already known lower bounds on the minimum distance of the previously known quasi-cyclic codes. Even though these codes do not improve the minimum distance of the best unstructured code known, their beautiful structure and simplicity provide several advantages in comparison to other...

In this paper, we investigate cyclic codes over the ring Fp[u, v, w]〈u , v, w, uv− vu, vw − wv, uw − wu〉, where p is a prime number. Which is a part of family of Frobenius rings. We find a unique set of generators for these codes and characterize the free cyclic codes. We also study the rank and the Hamming distance of these codes. We also constructs some good p−ary codes as the Gray images of ...

For any prime p, λ-constacyclic codes of length p over R = Fpm + uFpm are precisely the ideals of the local ring Rλ = R[x] 〈xp−λ〉 , where u = 0. In this paper, we first investigate the Hamming distances of cyclic codes of length p over R. The minimum Hamming distances of all cyclic codes of length p over R are determined. Moreover, an isometry between cyclic and α-constacyclic codes of length p...

In this paper, a class of shortened cyclic codes for a compound channel will be presented. This class of cyclic codes can be used for combating noises in a practical channel, where both burst errors and random errors occur. As compared with the results of a previous paper by Hsu, Kasami, and Chien (1968) (henceforward will be referred to as HKC codes), the shortened cyclic codes presented will ...