×

Matrix-product structure of constacyclic codes over finite chain rings \(\mathbb{F}_{p^m}[u]/\langle u^e\rangle\). (English) Zbl 1404.94145

Summary: Let \(m\), \(e\) be positive integers, \(p\) a prime number, \(\mathbb{F}_{p^m}\) be a finite field of \(p^m\) elements and \(R=\mathbb{F}_{p^m}[u]/\langle u^e\rangle\) which is a finite chain ring. For any \(\omega \in R^\times\) and positive integers \(k\), \(n\) satisfying \(\gcd(p,n)=1\), we prove that any \((1+\omega u)\)-constacyclic code of length \(p^kn\) over \(R\) is monomially equivalent to a matrix-product code of a nested sequence of \(p^k\) cyclic codes with length \(n\) over \(R\) and a \(p^k\times p^k\) matrix \(A_{p^k}\) over \(\mathbb{F}_p\). Using the matrix-product structures, we give an iterative construction of every \((1+\omega u)\)-constacyclic code by \((1+\omega u)\)-constacyclic codes of shorter lengths over \(R\).

MSC:

94B15 Cyclic codes
94B05 Linear codes (general theory)
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abualrub, T.; Siap, I., Constacyclic codes over \(\mathbb{F}_2+u\mathbb{F}_2\), J. Frankl. Inst., 346, 520-529, (2009) · Zbl 1176.94074 · doi:10.1016/j.jfranklin.2009.02.001
[2] Blackford, T., Negacyclic codes over \(Z_4\) of even length, IEEE Trans. Inf. Theory, 49, 1417-1424, (2003) · Zbl 1063.94101 · doi:10.1109/TIT.2003.811915
[3] Blockmore, T.; Norton, GH, Matrix-product codes over \(\mathbb{F}_q\), Appl. Algebra Eng. Commun. Comput., 12, 477-500, (2001) · Zbl 1004.94034 · doi:10.1007/PL00004226
[4] Cao, Y., On constacyclic codes over finite chain rings, Finite Fields Appl., 24, 124-135, (2013) · Zbl 1305.94097 · doi:10.1016/j.ffa.2013.07.001
[5] Cao, Y.; Cao, Y.; Fu, F-W, Cyclic codes over \(\mathbb{F}_{2^m}[u]/〈 u^k〉 \) of oddly even length, Appl. Algebra Eng. Commun. Comput., 27, 259-277, (2016) · Zbl 1344.94090 · doi:10.1007/s00200-015-0281-4
[6] Cao, Y.; Cao, Y.; Dong, L., Complete classification of \((δ + α u^2)\)-constacyclic codes over \(\mathbb{F}_{3^m}[u]/〈 u^4〉 \) of length \(3n\), Appl. Algebra Eng. Commun. Comput., 29, 13-39, (2018) · Zbl 1402.94099 · doi:10.1007/s00200-017-0328-9
[7] Cao, Y.; Cao, Y., The Gray image of constacyclic codes over the finite chain ring \(F_{p^m}[u]/〈 u^k〉 \), J. Appl. Math. Comput., (2017) · Zbl 1422.94044 · doi:10.1007/s12190-017-1107-2
[8] Cao, Y., Cao, Y., Dinh, H. Q., Fu, F-W., Gao, J., Sriboonchitta, S.: Constacyclic codes of length \(np^s\) over \({\mathbb{F}}_{p^m} + u{\mathbb{F}}_{p^m} \), https://www.researchgate.net/publication/320734899, October 2017. Accepted for publication in Adv. Math. Commun. · Zbl 1414.94933
[9] Dinh, HQ, On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions, Finite Fields Appl., 14, 22-40, (2008) · Zbl 1129.94047 · doi:10.1016/j.ffa.2007.07.001
[10] Dinh, HQ; Dhompongsa, S.; Sriboonchitta, S., Repeated-root constacyclic codes of prime power length over \(\frac{\mathbb{F}_{p^m}[u]}{〈 u^a〉 }\) and their duals, Discrete Math., 339, 1706-1715, (2016) · Zbl 1338.94104 · doi:10.1016/j.disc.2016.01.020
[11] Fan, Y.; Ling, S.; Liu, H., Matrix product codes over finite commutative Frobenius rings, Des. Codes Cryptogr., 71, 201-227, (2014) · Zbl 1342.94113 · doi:10.1007/s10623-012-9726-y
[12] Hernando, F.; Lally, K.; Ruano, D., Construction and decoding of matrix-product codes from nested codes, Appl. Algebra Eng. Commun. Comput., 20, 497-507, (2009) · Zbl 1178.94225 · doi:10.1007/s00200-009-0113-5
[13] Huffman, W.C., Pless, V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003) · Zbl 1191.94107 · doi:10.1017/CBO9780511807077
[14] Kai, X.; Zhu, S.; Li, P., \((1+λ u)\)-constacyclic codes over \(\mathbb{F}_p[u]/〈 u^k〉 \), J. Frankl. Inst., 347, 751-762, (2010) · Zbl 1286.94106 · doi:10.1016/j.jfranklin.2010.02.003
[15] Norton, G.; Sălăgean-Mandache, A., On the structure of linear and cyclic codes over finite chain rings, Appl. Algebra Eng. Commun. Comput., 10, 489-506, (2000) · Zbl 0963.94042 · doi:10.1007/PL00012382
[16] Özbudak, F.; Stichtenoth, H., Note on Niederreiter-Xing’s propagation rule for linear codes, Appl. Algebra Eng. Commun. Comput., 13, 53-56, (2002) · Zbl 0993.94556 · doi:10.1007/s002000100091
[17] Sobhani, R., Matrix-product structure of repeated-root cyclic codes over finite fields, Finite Fields Appl., 39, 216-232, (2016) · Zbl 1339.94091 · doi:10.1016/j.ffa.2016.02.001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.