Abstract
Let \(q\) be a power of a prime integer \(p, m=p^em_0\) and \(|q|_{m_{0}}\) the order of \(q\) modulo \(m_0\). By use of finite commutative chain ring theory, an algorithm to construct all distinct 1-generator quasi-cyclic codes with a fixed parity check polynomial over a finite field \(F_q\) of length \(mn\) and index \(n\), under the condition that \(\mathrm {gcd}(|q|_{m_0},n)=1\), are given.
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References
Bhargava V.K., Séguin G.E., Stein J.M.: Some \((mk, k)\) cyclic codes in quasi-cyclic form. IEEE Trans. Inf. Theory IT 24, 630–632 (1978).
Cao Y.: Structural properties and enumeration of 1-generator generalized quasi-cyclic codes. Des. Codes Cryptogr. 60, 67–79 (2011).
Cui J., Junying P.: Quaternary 1-generator quasi-cyclic codes. Des. Codes Cryptogr. 58, 23–33 (2011).
Dinh H.Q., López-permouth S.R.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans. Inf. Theory 50, 1728–1744 (2004).
Gulliver T.A., Bhargava V.K.: Two new rate 2/P binary quasi-cyclic codes. IEEE Trans. Inf. Theory 40, 1667–1668 (1994).
Hou X.-D., Leung K.H., Ma S.L.: On the groups of units of finite commutative chain rings. Finite Fields Appl. 9, 20–38 (2003).
Lally K.: Quasicyclic codes of index \(l\) over \(F_{q}\) viewed as \(F_{q}[x]\)-submodules of \(F_{q^{l}}[x]/\langle x^{m}-1 \rangle \). AAECC-15, Lecturer Notes in Computer Science, Vol. 2643 (pp. 244–253). Berlin: Springer (2003).
Ling S., Solé P.: On the algebraic structure of quasi-cyclic codes I: finite fields. IEEE Trans. Inf. Theory 47, 2751–2760 (2001).
Ling S., Solé P.: On the algebraic structure of quasi-cyclic codes II: chain rings. Des. Codes Cryptogr. 30, 113–130 (2003).
Ling S., Niederreiter H., Solé P.: On the algebraic structure of quasi-cyclic codes IV: repeated roots. Des. Codes Cryptogr. 38, 337–361 (2006).
Norton G., Sălăgean A.: On the structure of linear and cyclic codes over finite chain rings. Appl. Algebra Eng. Commun. Comput. 10, 489–506 (2000).
Séguin G.E.: A class of 1-generator quasi-cyclic codes. IEEE Trans. Inf. Theory 50, 1745–1753 (2004).
Tavares S.E., Bhargava V.K., Shiva S.G.S.: Some rate P/P+1 quasi-cyclic codes. IEEE Trans. Inf. Theory IT 20, 133–135 (1974).
Acknowledgments
This research is supported in part by the National Natural Science Foundation of China (No. 10971160).
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Communicated by J. Bierbrauer.
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Cao, Y. A class of 1-generator repeated root quasi-cyclic codes. Des. Codes Cryptogr. 72, 483–496 (2014). https://doi.org/10.1007/s10623-012-9777-0
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DOI: https://doi.org/10.1007/s10623-012-9777-0
Keywords
- 1-Generator repeated root quasi-cyclic code
- Finite commutative chain ring
- Parity check polynomial
- Group of units