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.
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The authors would like to express their sincere gratitude to the referees for carefully reading the manuscript and for their suggestions which improved the readability of the text.
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Communicated by G. Korchmaros.
Prof. Dr. S. M. Dodunekov sadly, and totally unexpectedly, passed away in August 2012.
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van Zanten, A.J., Bojilov, A. & Dodunekov, S.M. Generalized residue and t-residue codes and their idempotent generators. Des. Codes Cryptogr. 75, 315–334 (2015). https://doi.org/10.1007/s10623-013-9905-5
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DOI: https://doi.org/10.1007/s10623-013-9905-5
Keywords
- Cyclic codes
- Generalized quadratic residue codes
- \(t\)-Residue codes
- Cubic residue codes
- Idempotent generators
- Generator polynomials
- Group rings