Quadratic-Residue Codes and Cyclotomic Fields

Abstract

We present an exposition of quadratic-residue codes through their embedding in codes over the quadratic subfield of the \(p\)th cyclotomic field, the algebraic number field of \(p\)th roots of unity. This representation allows the development of effective syndrome decoding algorithms that can fully exploit the code’s error-correcting capability. This is accomplished via Galois automorphisms of cyclotomic fields. For each fixed \(p\), the general results hold for all pairs \((p,q)\), with a finite number of exceptions that depends on \(p\). A complete discussion of the set of quadratic-residue codes of length \(23\) and dimension \(12\) illustrates these results. This set includes the Golay code, the only perfect binary three-error-correcting code.