Generalizing binary quadratic residue codes to higher power residues over larger fields
The binary quadratic residue codes are well-known codes in coding theory. Their construction starts from \(p\equiv \pm1\pmod{8}\) so that 2 is a quadratic residue modulo \(p\). This article investigates a generalization of these quadratic residue codes. Let \(p\) and \(q\) be two distinct prime numbers such that \(q|(p-1)\) and such that \(q\) is a \(q\)-th power residue modulo \(p\). Then there exists an element \(\beta\in\mathbb{F}_p\) such that \(\beta^q\equiv q \pmod{p}\), and it is possible to divide \(\mathbb{F}_p\) into \(q\) cosets. These cosets are the basis to define the \(q\)-th power residue codes. The author presents generating polynomials for these codes, and defines a new notion corresponding to the binary concept of an idempotent. The construction of these \(q\)-th power residue codes is also briefly explained in Section 15.2 of \textit{E. W. Berlekamp} [Algebraic Coding Theory (McGraw-Hill, New York, NY) (1968; Zbl 0988.94521)]. Using different techniques from the book of Berlekamp, the author presents a lower bound on the codeword weight of the dual of these \(q\)-th power residue codes. This lower bound then leads to a lower bound on the weight of the codewords of the \(q\)-th power residue codes.
scientific article; zbMATH DE number 5554720