Quasi-cyclic subcodes of cyclic codes

TL;DR

This paper characterizes the possible indices of quasi-cyclic subcodes within cyclic codes, providing enumeration results and linking the problem to vector subspace enumeration in finite fields, with applications to simplex and BCH codes.

Contribution

It offers a complete characterization of quasi-cyclic subcode indices and connects enumeration to finite field vector subspaces, advancing understanding of cyclic code structures.

Findings

Enumeration of quasi-cyclic subcodes for specific codes

Connection between subcode enumeration and finite field vector spaces

Application to simplex and BCH codes

Abstract

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 simplex code and duals of certain BCH codes. Our results are based on the trace representation of cyclic codes.