On self-dual cyclic codes and generalized self-dual cyclic codes over finite fields - Yan, Jia (2012)
Cyclic codes over finite fields form an important class of codes which has been extensively studied, for their interesting algebraic structure, and many “good” codes are cyclic. Therefore, it is natural to try to generalize the notion, for example in the search of good codes, or to generalize the algebraic properties. The duality of cyclic codes and the algebraic structure of generalized cyclic codes are the main objects of this work. A cyclic code over a finite field is a vector space over a finite field, and the dual is the orthogonal complement. When this dual code is the same as the original code, we call the code self-dual. When the dual code can be obtained by a certain type of transformation from the original code, then we call the code isodual (“isomorphic” to dual in some sense).For self-duality, it is shown that self-dual cyclic codes of length n over Fq exist if and only if n is even and q = 2m with m a positive integer. The enumeration of such codes is also investigated. When n and q are even, there is always a trivial self-dual cyclic code with generator polynomial x n2 + 1. We therefore classify the existence of self-dual cyclic codes, for given n and q, into two cases: when only the trivial one exists and when two or more such codes exist. Given n and m, an easy criterion to determine which of these two cases occurs is given in terms of the prime factors of n, for most n. It is also shown that, over a fixed field, the latter case occurs more frequently as the length grows. For isoduality, two classes of isodual cyclic codes are considered: cyclic codes equivalent to their dual codes up to a multiplier permutation and cyclic codes equivalent to their dual codes up to a scalar transformation. The construction as well as the enumeration are given. Cyclic codes are namely invariant under cyclic shift on their codewords. To generalize the notion, we can define quasi-cyclic codes: the codes that are invariant under l-cyclic shifts on their codewords, for some positive integer l. We can also define constacyclic codes: the codes that are invariant under cyclic shift while multiplying the shifted part of the codewords by a constant. Combining the above two notions, we have quasi-twisted (QT) codes. For generalized cyclic codes, we mainly focus on QT codes since QT codes include cyclic codes, quasi-cyclic codes and constacyclic codes as special cases. We decompose a QT code to a direct sum of component codes – linear codes over rings. Furthermore, given the decomposition of a QT code, we can describe the decomposition of its dual code. We also use the generalized discrete Fourier transform to give the inverse formula for both the so-called nonrepeated-root and so-called repeated-root cases. Then we produce a formula which can be used to construct a QT code given the component codes.