On Expanded Cyclic Codes

TL;DR

This paper provides an explicit construction of expanded cyclic codes, characterizes their properties, identifies constant-weight subclasses, and analyzes their asymptotic performance, revealing limitations of binary expanded Reed-Solomon codes.

Contribution

It introduces explicit descriptions of expanded cyclic codes, characterizes their structure, and challenges assumptions about their asymptotic goodness.

Findings

Expanded cyclic codes maintain algebraic structure with explicit matrices.

Certain BCH codes are constant-weight and achieve the Plotkin bound.

Binary expanded Reed-Solomon codes are asymptotically

Abstract

The paper has a threefold purpose. The first purpose is to present an explicit description of expanded cyclic codes defined in $\GF(q^m)$. The proposed explicit construction of expanded generator matrix and expanded parity check matrix maintains the symbol-wise algebraic structure and thus keeps many important original characteristics. The second purpose of this paper is to identify a class of constant-weight cyclic codes. Specifically, we show that a well-known class of $q$-ary BCH codes excluding the all-zero codeword are constant-weight cyclic codes. Moreover, we show this class of codes achieve the Plotkin bound. The last purpose of the paper is to characterize expanded cyclic codes utilizing the proposed expanded generator matrix and parity check matrix. We characterize the properties of component codewords of a codeword and particularly identify the precise conditions under which a codeword can be represented by a subbasis. Our developments reveal an alternative while more general view on the subspace subcodes of Reed-Solomon codes. With the new insights, we present an improved lower bound on the minimum distance of an expanded cyclic code by exploiting the generalized concatenated structure. We also show that the fixed-rate binary expanded Reed-Solomon codes are asymptotically "bad", in the sense that the ratio of minimum distance over code length diminishes with code length going to infinity. It overturns the prevalent conjecture that they are "good" codes and deviates from the ensemble of generalized Reed-Solomon codes which asymptotically achieves the Gilbert-Varshamov bound.