Generalized residue and t-residue codes and their idempotent generators

Abstract

In this paper a general class of linear cyclic codes $C_{n,q,t}^i , 1\le i\le t$, is defined of length $n$ and over a field ${ GF}(q)$ with $(n,q)=1$. This class of codes includes as special cases quadratic residue codes, generalized quadratic residue codes, $e$-residue codes and $Q$-codes. Furthermore, they partially overlap with the families of duadic, triadic and polyadic codes. Expressions for idempotent generators are derived in terms of the size of cyclotomic cosets mod $n$ and coefficients of the irreducible polynomials over ${ GF}(q)$ dividing $x^{n}-1$. As an auxiliary tool an orthonormal matrix is introduced whose columns correspond to these idempotents. Concrete examples are presented for $t=2$ and $n\in \{p^{\lambda },2p^{\lambda },2^{\lambda }\}, \lambda \ge 1$, where $p$ is an arbitrary odd prime. When $n=p^{\lambda }$ or $n=2p^{\lambda }$ the codes all belong to the subclass of 2-residue codes. Using this technique, we determine the idempotents of the codes $C_{2^{\lambda },q,2}^i$, and recover those of the generalized quadratic codes $C_{p^{\lambda },q,2}^i$ and of the codes $C_{2p^{\lambda },q,2}^i$. In the final section the idempotents of the cubic residue codes $C_{p,q,3}^i$ are constructed.