Abstract
Constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, (1 − uv)-constacyclic codes over the local ring \(\mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p \) are studied. It is proved that the image of a (1 − uv)-constacyclic code of length n over \(\mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p \) under a Gray map is a distance invariant quasi-cyclic code of index p 2 and length p 3 n over \(\mathbb{F}_p \). Several examples of optimal linear codes over \(\mathbb{F}_p \) from (1 − uv)-constacyclic codes over \(\mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p \) are given.
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References
Wolfmann J, Negacyclic and cyclic codes over ℤ4, IEEE Trans. Inform. Theory, 1999, 45(7): 2527–2532.
Wolfmann J, Binary images of cyclic codes over ℤ4, IEEE Trans. Inform. Theory, 2001, 47(5): 1773–1779.
Ling S and Blackford J T, \(\mathbb{Z}_{p^{k + 1} } \)-linear codes, IEEE Trans. Inform. Theory, 2002, 48(9): 2592–2605.
Abualrub T and Siap T, Constacyclic codes over \(\mathbb{F}_2 + u\mathbb{F}_2 \), J. Franklin Inst., 2009, 346(5): 520–529.
Amarra M C V and Nemenzo F R, On (1−u)-cyclic codes over \(\mathbb{F}_{p^k } + u\mathbb{F}_{p^k } \), Applied Mathematics Letters, 2008, 21(11): 1129–1133.
Dinh H Q and López-Permouth S R, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 2004, 50(8): 1728–1744.
Kai X S, Zhu S X, and Li P, (1+λu)-Constacyclic codes over \(\mathbb{F}_p [u]/\left\langle {u^m } \right\rangle \), J. Franklin Inst., 2010, 347(5): 751–762.
Yildiz B and Karadenniz S, Linear codes over \(\mathbb{F}_2 + u\mathbb{F}_2 + v\mathbb{F}_2 + uv\mathbb{F}_2 \), Des. Codes Cryptogr., 2010, 54(1): 61–81.
Yildiz B and Karadenniz S, Self-dual codes over \(\mathbb{F}_2 + u\mathbb{F}_2 + v\mathbb{F}_2 + uv\mathbb{F}_2 \), J. Franklin Inst., 2010, 347(10): 1888–1894.
Yildiz B and Karadenniz S, Cyclic codes over \(\mathbb{F}_2 + u\mathbb{F}_2 + v\mathbb{F}_2 + uv\mathbb{F}_2 \), Des. Codes Cryptogr., 2011, 58(3): 221–234.
Karadenniz S and Yildiz B, (1 + v)-Constacyclic codes over \(\mathbb{F}_2 + u\mathbb{F}_2 + v\mathbb{F}_2 + uv\mathbb{F}_2 \), J. Franklin Inst., 2011, 348(9): 2625–2632.
Kai X S and Zhu S X, A family of constacyclic codes over \(\mathbb{F}_2 + u\mathbb{F}_2 + v\mathbb{F}_2 + uv\mathbb{F}_2 \), Journal of Systems Science and Complexity, 2012, 25(5): 1032–1040.
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This research was supported by the National Natural Science Foundation of China under Grant No. 61370089, the Natural Science Foundation of Anhui Province under Grant No. 1208085MA14, the Natural Science Fund of Education Department of Anhui province under Grant No. KJ2013Z276, the Fundamental Research Funds of Hefei University under Grant No. 10KY01ZD, and the Key construction discipline Funds of Hefei University under Grant No. 2014XK08.
This paper was recommended for publication by Editor HU Lei.
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Yu, H., Zhu, S. & Kai, X. (1 − uv)-constacyclic codes over \(\mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p \) . J Syst Sci Complex 27, 811–816 (2014). https://doi.org/10.1007/s11424-014-3241-3
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DOI: https://doi.org/10.1007/s11424-014-3241-3