SIMPLE-ROOT NEGACYCLIC CODES OF LENGTH 2pnm OVER A FINITE FIELD

Title & Authors
SIMPLE-ROOT NEGACYCLIC CODES OF LENGTH 2pnm OVER A FINITE FIELD

Abstract
Let p, $\small{{\ell}}$ be distinct odd primes, q be an odd prime power with gcd(q, p) = gcd(q,$\small{{\ell}}$) = 1, and m, n be positive integers. In this paper, we determine all self-dual, self-orthogonal and complementary-dual negacyclic codes of length $\small{2p^{n{\ell}m}}$ over the finite field $\small{{\mathbb{F}}_q}$ with q elements. We also illustrate our results with some examples.
Keywords
dual code;constacyclic codes;cyclotomic cosets;
Language
English
Cited by
1.
On constacyclic codes over finite fields, Cryptography and Communications, 2016, 8, 4, 617
References
1.
T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976.

2.
G. K. Bakshi and M. Raka, A class of constacyclic codes over a finite field, Finite Fields Appl. 18 (2012), no. 2, 362-377.

3.
G. K. Bakshi and M. Raka, Self-dual and self-orthogonal negacyclic codes of length $2p^n$ over a finite field, Finite Fields Appl. 19 (2013), no. 1, 39-54.

4.
E. R. Berlekamp, Negacyclic codes for the Lee metric, Proc. Combin. Math. Appl., 298-316, Univ. North Carolina Press, Chapel Hill, 1969.

5.
E. R. Berlekamp, Algebraic Coding Theory, McGraw -Hill Book Company, New York, 1968.

6.
T. Blackford, Negacyclic duadic codes, Finite Fields Appl. 14 (2008), no. 4, 930-943.

7.
H. Q. Dinh, Constacyclic codes of length $p^s$ over ${\mathbb{F}}_{p^m}$ + $u{\mathbb{F}}_{p^m}$, J. Algebra 324 (2010), no. 5, 940-950.

8.
H. Q. Dinh, Repeated-root constacyclic codes of length $2p^s$, Finite Fields Appl. 18 (2012), no. 1, 133-143.

9.
H. Q. Dinh, Structure of repeated-root constacyclic codes of length $3p^s$ and their duals, Discrete Math. 313 (2013), no. 9, 983-991.

10.
H. Q. Dinh, On repeated-root constacyclic codes of length $4p^s$, Asian-Eur. J. Math. 6 (2013), no. 2, 1350020, 25 pp.

11.
H. Q. Dinh, Repeated-root cyclic and negacyclic codes of length $6p^s$, Ring theory and its applications, 69-87, Contemp. Math., 609, Amer. Math. Soc., Providence, RI, 2014.

12.
K. Guenda and T. A. Gulliver, Self-dual repeated-root cyclic and negacyclic codes over finite fields, Proc. IEEE Int. Symp. Inform. Theory 2012 (2012), 2904-2908.

13.
Y. Jia, S. Ling, and C. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inform. Theory 57 (1994), no. 4, 2241-2251.

14.
R. G. Kelsch and D. H. Green, Nonbinary negacyclic code which exceeds Berlekamp's (p-1)/2 bound, Electron. Lett. 7 (1971), 664-665.

15.
J. L. Massey, Linear codes with complementary duals, Discrete Math. 106/107 (1992), 337-342.

16.
E. Prange, Cyclic error-correcting codes in two symbols, Air Force Cambridge Research Labs, Bedford, Mass, TN-57-103, 1957.

17.
N. Sendrier, Linear codes with complementary duals meet the Gilbert-Varshamov bound, Discrete Math. 285 (2004), no. 1-3, 345-347.

18.
A. Sharma, Constacyclic codes over finite fields, Communicated for publication.

19.
A. Sharma, Self-dual and self-orthogonal negacyclic codes of length $2^mp^n$ over a finite field, Discrete Math. 338 (2015), no. 4, 576-592.

20.
A. Sharma, Self-orthogonal and complementary-dual cyclic codes of length $p^n{\el}^m$ over a finite field, Communicated for publication.

21.
A. Sharma, Repeated-root constacyclic codes of length ${\el}^tp^s$ and their dual codes, Cryptogr. Commun. 7 (2015), no. 2, 229-255.

22.
A. Sharma, G. K. Bakshi, and M. Raka, Polyadic codes of prime power length, Finite Fields Appl. 13 (2007), no. 4, 1071-1085.

23.
X. Yang and J. L. Massey, The condition for a cyclic code to have a complementary dual, Discrete Math. 126 (1994), no. 1-3, 391-393.