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

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

Ekkasit Sangwisut , Somphong Jitman 2,, and  Patanee Udomkavanich 3,

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.

Keywords: Constacyclic codes, quasi-twisted codes, Hermitian inner product, self-dual codes, complementary dual cods.
Mathematics Subject Classification: Primary: 94B15, 11T06; Secondary: 94B05.
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:
[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.

[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.

[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.

[4]

E. R. Berlekamp, Algebraic Coding Theory McGraw-Hill, New York, 1968.

[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.

[6]

B. Chen, Y. Fan, L. Lin and H. Liu, Constacyclic codes over finite fields, Finite Fields Appl., 18 (2012), 1217-1231. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

T. Kasami, A Gilbert-Varshamov bound for quasi-cyclic codes of rate $1/2$ IEEE Trans. Inf. Theory 20 (1974), 679.

[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.

[17]

G. G. La Guardia On optimal constacyclic codes preprint, arXiv: 1311.2505 doi: 10.1016/j.laa.2016.02.014.

[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.

[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.

[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.

[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.

[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.

[23]

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory Springer-Verlag, Berlin, 2006.

[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.

[25]

E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory, Elsevier, North-Holland, 1998,177–294.

[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.

[27]

J. P. Serre, A Course in Arithmetic Springer-Verlag, New York, 1973.

[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.

[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.

show all references

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.

[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.

[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.

[4]

E. R. Berlekamp, Algebraic Coding Theory McGraw-Hill, New York, 1968.

[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.

[6]

B. Chen, Y. Fan, L. Lin and H. Liu, Constacyclic codes over finite fields, Finite Fields Appl., 18 (2012), 1217-1231. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

T. Kasami, A Gilbert-Varshamov bound for quasi-cyclic codes of rate $1/2$ IEEE Trans. Inf. Theory 20 (1974), 679.

[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.

[17]

G. G. La Guardia On optimal constacyclic codes preprint, arXiv: 1311.2505 doi: 10.1016/j.laa.2016.02.014.

[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.

[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.

[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.

[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.

[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.

[23]

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory Springer-Verlag, Berlin, 2006.

[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.

[25]

E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory, Elsevier, North-Holland, 1998,177–294.

[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.

[27]

J. P. Serre, A Course in Arithmetic Springer-Verlag, New York, 1973.

[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.

[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.

Table 1.  MDS constacyclic Hermitian self-dual codes over $\mathbb{F}_{q^2}$
$q$ $m$ $i$ Parameters $T$
$3$ $2$ $1$ $[4,2,3]$ $\{1, 3\}$
$5$ $1$ $1$ $[6,3,4]$ $\{1, 3, 5\}$
$7$ $3$ $1$ $[8,4,5]$ $\{1, 3, 5, 7\}$
$3$ $2$ $[4,2,3]$ $\{1, 5\}$
$9$ $1$ $1$ $[10,5,6]$ $\{1, 3, 5, 7, 9\}$
$11$ $2$ $1$ $[12,6,7]$ $\{1, 3, 5, 7, 9, 11\}$
$2$ $2$ $[6,3,4]$ $\{1, 5, 9\}$
$13$ $1$ $1$ $[14,7,8]$ $\{1, 3, 5, 7, 9, 11, 13\}$
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$q$ $m$ $i$ 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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