Minimal quadratic residue cyclic codes of length \(p^n\) (\(p\) odd prime) (Q2761611)

A cyclic code of length \(p^n\) over the field \(F\) \((p\) prime, \(F\) a finite field with \(q\) elements where \((p,q)=1)\) can be viewed as an ideal in the semisimple ring \(F[X]/ \langle X^{p^n}-1\rangle= R_{p^n}\). Cyclic codes over \(F\) are quadratic residue codes. In this paper the minimal quadratic residue codes of length \(p^n\) \((p\) odd prime) over \(F\) are described. The explicit expressions for the \(2n+1\) primitive idempotents in \(R_{p^n}\) are obtained. The generating polynomials of the codes generated by these primitive idempotents are optimal. As to the purpose of computing the generating polynomials numerically, an algorithm is easy to produce. A lot of worked-out examples close the paper on that score. The reviewer considers this paper as a handy source for educational purposes in coding theory.