American Institute of Mathematical Sciences

August  2017, 11(3): 595-613. doi: 10.3934/amc.2017045

Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields

 1 Department of Mathematics and Statistics, Faculty of Science, Thaksin University, Phatthalung Campus, Phatthalung 93110, Thailand 2 Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand 3 Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

* Corresponding author

Received  November 2015 Published  August 2017

Fund Project: S. Jitman was supported by the Thailand Research Fund under Research Grant TRG5780065

Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields are studied. An alternative algorithm for factorizing $x^n-\lambda$ over ${\mathbb{F}_{{q^2}}}$ is given, where $λ$ is a unit in ${\mathbb{F}_{{q^2}}}$. Based on this factorization, the dimensions of the Hermitian hulls of $\lambda$-constacyclic codes of length $n$ over ${\mathbb{F}_{{q^2}}}$ are determined. The characterization and enumeration of constacyclic Hermitian self-dual (resp., complementary dual) codes of length $n$ over ${\mathbb{F}_{{q^2}}}$ are given through their Hermitian hulls. Subsequently, a new family of MDS constacyclic Hermitian self-dual codes over ${\mathbb{F}_{{q^2}}}$ is introduced.

As a generalization of constacyclic codes, quasi-twisted Hermitian self-dual codes are studied. Using the factorization of $x^n-\lambda$ and the Chinese Remainder Theorem, quasi-twisted codes can be viewed as a product of linear codes of shorter length over some extension fields of ${\mathbb{F}_{{q^2}}}$. Necessary and sufficient conditions for quasi-twisted codes to be Hermitian self-dual are given. The enumeration of such self-dual codes is determined as well.

Citation: Ekkasit Sangwisut, Somphong Jitman, Patanee Udomkavanich. Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields. Advances in Mathematics of Communications, 2017, 11 (3) : 595-613. doi: 10.3934/amc.2017045
References:
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References:
 [1] N. Aydin, T. Asamov and T. A. Gulliver, Some open problems on quasi-twisted and related code constructions and good quaternary codes, in Proc. IEEE ISIT' 2007, Nice, France, 2007, 856–860.Google Scholar [2] N. Aydin, I. Siap and D. J. Ray-Chaudhuri, The structure of 1-generator quasi-twisted codes and new linear codes, Des. Codes Crypt., 24 (2001), 313-326. doi: 10.1023/A:1011283523000. Google Scholar [3] G. K. Bakshi and M. Raka, A class of constacyclic codes over a finite field, Finite Fields Appl., 18 (2012), 362-377. doi: 10.1016/j.ffa.2011.09.005. Google Scholar [4] E. R. Berlekamp, Algebraic Coding Theory McGraw-Hill, New York, 1968. Google Scholar [5] B. Chen, H. Q. Dinh, Y. Fan and S. Ling, Polyadic constacyclic codes, IEEE Trans. Inf. Theory, 61 (2015), 1217-1231. doi: 10.1109/TIT.2015.2451656. Google Scholar [6] B. Chen, Y. Fan, L. Lin and H. Liu, Constacyclic codes over finite fields, Finite Fields Appl., 18 (2012), 1217-1231. Google Scholar [7] E. Z. Chen, An explicit construction of 2-generator quasi-twisted codes, IEEE Trans. Inf. Theory, 54 (2008), 5770-5773. doi: 10.1109/TIT.2008.2006430. Google Scholar [8] V. Chepyzhov, A Gilbert-Vashamov bound for quasitwisted codes of rate $\frac1n$, in Proc. Joint Swedish-Russian Int. Workshop Inf. Theory, Mölle, Sweden, 1993,214–218.Google Scholar [9] R. Daskalov and P. Hristov, New quasi-twisted degenerate ternary linear codes, IEEE Trans. Inf. Theory, 49 (2003), 2259-2263. doi: 10.1109/TIT.2003.815798. Google Scholar [10] T. A. Gulliver, M. Harada and H. Miyabayashi, Double circulant and quasi-twisted self-dual codes over $\mathbb F_5$ and $\mathbb F_7$, Adv. Math. Commun., 1 (2007), 223-238. doi: 10.3934/amc.2007.1.223. Google Scholar [11] T. A. Gulliver, N. P. Secord and S. A. Mahmoud, A link between quasi-cyclic codes and convolution codes, IEEE Trans. Inf. Theory, 44 (1998), 431-435. doi: 10.1109/18.651076. Google Scholar [12] Y. Jia, On quasi-twisted codes over finite fields, Finite Fields Appl., 18 (2012), 237-257. doi: 10.1016/j.ffa.2011.08.001. Google Scholar [13] Y. Jia, S. Ling and C. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inf. Theory, 13 (2011), 2243-2251. doi: 10.1109/TIT.2010.2092415. Google Scholar [14] X. Kai, X. Zhu and P. Li, Constacyclic codes and some new quantum MDS codes, IEEE Trans. Inf. Theory, 60 (2014), 2080-2086. doi: 10.1109/TIT.2014.2308180. Google Scholar [15] T. Kasami, A Gilbert-Varshamov bound for quasi-cyclic codes of rate $1/2$ IEEE Trans. Inf. Theory 20 (1974), 679. Google Scholar [16] A. Ketkar, A. Klappenecker, S. Kumar and P. K. Sarvepalli, Nonbinary stabilizer codes over finite fields, IEEE Trans. Inf. Theory, 52 (2006), 4892-4914. doi: 10.1109/TIT.2006.883612. Google Scholar [17] G. G. La Guardia On optimal constacyclic codes preprint, arXiv: 1311.2505 doi: 10.1016/j.laa.2016.02.014. Google Scholar [18] L. Lin, H. Liu and B. Chen, Existence conditions for self-orthogonal negacyclic codes over finite fields, Adv. Math. Commun., 9 (2015), 1-7. doi: 10.3934/amc.2015.9.1. Google Scholar [19] S. Ling, H. Niederreiter and P. Solé, On the algebraic structure of quasi-cyclic codes Ⅳ: Repeated roots, Des. Codes Crypt., 38 (2006), 337-361. doi: 10.1007/s10623-005-1431-7. Google Scholar [20] S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes Ⅰ: Finite fields, IEEE Trans. Inf. Theory, 47 (2001), 2751-2760. doi: 10.1109/18.959257. Google Scholar [21] S. Ling and P. Solé, Good self-dual quasi-cyclic codes exist, IEEE Trans. Inf. Theory, 49 (2003), 1052-1053. doi: 10.1109/TIT.2003.809501. Google Scholar [22] S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes Ⅲ: Generator theory, IEEE Trans. Inf. Theory, 51 (2005), 2692-2700. doi: 10.1109/TIT.2005.850142. Google Scholar [23] G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory Springer-Verlag, Berlin, 2006. Google Scholar [24] H. Özadamb and F. Özbudak, A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$, Adv. Math. Commun., 3 (2009), 265-271. doi: 10.3934/amc.2009.3.265. Google Scholar [25] E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory, Elsevier, North-Holland, 1998,177–294. Google Scholar [26] E. Sangwisut, S. Jitman, S. Ling and P. Udomkavanich, Hulls of cyclic and negacyclic codes over finite fields, Finite Fields Appl., 33 (2015), 232-257. doi: 10.1016/j.ffa.2014.12.008. Google Scholar [27] J. P. Serre, A Course in Arithmetic Springer-Verlag, New York, 1973. Google Scholar [28] G. Solomon and H. C. A. van Tilborg, A connection between block and convolutional codes, SIAM J. Appl. Math., 37 (1979), 358-369. doi: 10.1137/0137027. Google Scholar [29] Y. Yang and W. Cai, On self-dual constacyclic codes over finite fields, Des. Codes Crypt., 74 (2015), 355-364. doi: 10.1007/s10623-013-9865-9. Google Scholar
MDS constacyclic Hermitian self-dual codes over $\mathbb{F}_{q^2}$
 $q$ $m$ $i$ Parameters $T$ $3$ $2$ $1$ $[4,2,3]$ $\{1, 3\}$ $5$ $1$ $1$ $[6,3,4]$ $\{1, 3, 5\}$ $7$ $3$ $1$ $[8,4,5]$ $\{1, 3, 5, 7\}$ $3$ $2$ $[4,2,3]$ $\{1, 5\}$ $9$ $1$ $1$ $[10,5,6]$ $\{1, 3, 5, 7, 9\}$ $11$ $2$ $1$ $[12,6,7]$ $\{1, 3, 5, 7, 9, 11\}$ $2$ $2$ $[6,3,4]$ $\{1, 5, 9\}$ $13$ $1$ $1$ $[14,7,8]$ $\{1, 3, 5, 7, 9, 11, 13\}$
 $q$ $m$ $i$ Parameters $T$ $3$ $2$ $1$ $[4,2,3]$ $\{1, 3\}$ $5$ $1$ $1$ $[6,3,4]$ $\{1, 3, 5\}$ $7$ $3$ $1$ $[8,4,5]$ $\{1, 3, 5, 7\}$ $3$ $2$ $[4,2,3]$ $\{1, 5\}$ $9$ $1$ $1$ $[10,5,6]$ $\{1, 3, 5, 7, 9\}$ $11$ $2$ $1$ $[12,6,7]$ $\{1, 3, 5, 7, 9, 11\}$ $2$ $2$ $[6,3,4]$ $\{1, 5, 9\}$ $13$ $1$ $1$ $[14,7,8]$ $\{1, 3, 5, 7, 9, 11, 13\}$
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