2015, 9(4): 437-447. doi: 10.3934/amc.2015.9.437

Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields

1. 

Centre for Advanced Study in Mathematics, Panjab University, Chandigarh 160014, India, India

Received  October 2013 Revised  December 2014 Published  November 2015

The main objective of this article is to study self-orthogonal negacyclic codes of length $n$ over a finite field $\mathbb{F}_{q}$, where the characteristic of $\mathbb{F}_{q}$ does not divide $n$. We investigate issues related to their existence, characterization and enumeration. We find the necessary and sufficient conditions for the existence of self-orthogonal negacyclic codes of length $n$ over a finite field $\mathbb{F}_{q}$. We characterize the defining sets and the corresponding generator polynomials of these codes. We obtain formulae to calculate the number of self-dual and self-orthogonal negacyclic codes of a given length $n$ over $\mathbb{F}_{q}$. The enumeration formula for self-orthogonal negacyclic codes involves a two-variable function $\chi(d,q)$ defined by $\chi(d,q)=0$ if $d$ divides $(q^{k}+1)$ for some $k\geq0$ and $\chi(d,q)=1$, otherwise. We give necessary and sufficient conditions when $\chi(d,q)=0$ holds.
Citation: Amita Sahni, Poonam Trama Sehgal. Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (4) : 437-447. doi: 10.3934/amc.2015.9.437
References:
[1]

G. K. Bakshi and M. Raka, A class of constacyclic codes over a finite field,, Finite Fields Appl., 18 (2012), 362. doi: 10.1016/j.ffa.2011.09.005.

[2]

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), 39. doi: 10.1016/j.ffa.2012.10.003.

[3]

T. Blackford, Negacyclic duadic codes,, Finite Fields Appl., 14 (2008), 930. doi: 10.1016/j.ffa.2008.05.004.

[4]

H. Q. Dinh, Repeated-root constacyclic codes of length $2p^s$,, Finite Fields Appl., 18 (2012), 133. doi: 10.1016/j.ffa.2011.07.003.

[5]

H. Q. Dinh, Structure of repeated-root constacyclic codes of length $3p^s$ and their duals,, Discrete Mathematics, 313 (2013), 983. doi: 10.1016/j.disc.2013.01.024.

[6]

H. Q. Dinh, On repeated-root constacyclic codes of length $4p^s$,, Asian-European J. Math., 6 (2013). doi: 10.1142/S1793557113500204.

[7]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings,, IEEE Trans. Inform. Theory, 50 (2004), 1728. doi: 10.1109/TIT.2004.831789.

[8]

W. Cary Huffman and V. Pless, Fundamentals of Error-Correcting Codes,, Cambridge University Press, (2003). doi: 10.1017/CBO9780511807077.

[9]

Y. Jia, S. Ling and C. Xing, On self-dual cyclic codes over finite fields,, IEEE Trans. Inform. Theory, 57 (2011), 2243. doi: 10.1109/TIT.2010.2092415.

[10]

J. S. Leon and V. Pless, Self-dual codes over GF(5),, Journal of Combinatorial Theory, 32 (1982), 178. doi: 10.1016/0097-3165(82)90019-X.

[11]

F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes,, North-Holland, (1977).

[12]

V. Pless, A classification of self-orthogonal codes over GF(2),, Discrete Mathematics, 3 (1972), 209. doi: 10.1016/0012-365X(72)90034-9.

show all references

References:
[1]

G. K. Bakshi and M. Raka, A class of constacyclic codes over a finite field,, Finite Fields Appl., 18 (2012), 362. doi: 10.1016/j.ffa.2011.09.005.

[2]

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), 39. doi: 10.1016/j.ffa.2012.10.003.

[3]

T. Blackford, Negacyclic duadic codes,, Finite Fields Appl., 14 (2008), 930. doi: 10.1016/j.ffa.2008.05.004.

[4]

H. Q. Dinh, Repeated-root constacyclic codes of length $2p^s$,, Finite Fields Appl., 18 (2012), 133. doi: 10.1016/j.ffa.2011.07.003.

[5]

H. Q. Dinh, Structure of repeated-root constacyclic codes of length $3p^s$ and their duals,, Discrete Mathematics, 313 (2013), 983. doi: 10.1016/j.disc.2013.01.024.

[6]

H. Q. Dinh, On repeated-root constacyclic codes of length $4p^s$,, Asian-European J. Math., 6 (2013). doi: 10.1142/S1793557113500204.

[7]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings,, IEEE Trans. Inform. Theory, 50 (2004), 1728. doi: 10.1109/TIT.2004.831789.

[8]

W. Cary Huffman and V. Pless, Fundamentals of Error-Correcting Codes,, Cambridge University Press, (2003). doi: 10.1017/CBO9780511807077.

[9]

Y. Jia, S. Ling and C. Xing, On self-dual cyclic codes over finite fields,, IEEE Trans. Inform. Theory, 57 (2011), 2243. doi: 10.1109/TIT.2010.2092415.

[10]

J. S. Leon and V. Pless, Self-dual codes over GF(5),, Journal of Combinatorial Theory, 32 (1982), 178. doi: 10.1016/0097-3165(82)90019-X.

[11]

F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes,, North-Holland, (1977).

[12]

V. Pless, A classification of self-orthogonal codes over GF(2),, Discrete Mathematics, 3 (1972), 209. doi: 10.1016/0012-365X(72)90034-9.

[1]

Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031

[2]

Liren Lin, Hongwei Liu, Bocong Chen. Existence conditions for self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (1) : 1-7. doi: 10.3934/amc.2015.9.1

[3]

Masaaki Harada, Akihiro Munemasa. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6 (2) : 229-235. doi: 10.3934/amc.2012.6.229

[4]

Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267

[5]

Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047

[6]

Masaaki Harada, Akihiro Munemasa. On the covering radii of extremal doubly even self-dual codes. Advances in Mathematics of Communications, 2007, 1 (2) : 251-256. doi: 10.3934/amc.2007.1.251

[7]

Stefka Bouyuklieva, Iliya Bouyukliev. Classification of the extremal formally self-dual even codes of length 30. Advances in Mathematics of Communications, 2010, 4 (3) : 433-439. doi: 10.3934/amc.2010.4.433

[8]

Hyun Jin Kim, Heisook Lee, June Bok Lee, Yoonjin Lee. Construction of self-dual codes with an automorphism of order $p$. Advances in Mathematics of Communications, 2011, 5 (1) : 23-36. doi: 10.3934/amc.2011.5.23

[9]

Bram van Asch, Frans Martens. Lee weight enumerators of self-dual codes and theta functions. Advances in Mathematics of Communications, 2008, 2 (4) : 393-402. doi: 10.3934/amc.2008.2.393

[10]

Bram van Asch, Frans Martens. A note on the minimum Lee distance of certain self-dual modular codes. Advances in Mathematics of Communications, 2012, 6 (1) : 65-68. doi: 10.3934/amc.2012.6.65

[11]

Masaaki Harada, Katsushi Waki. New extremal formally self-dual even codes of length 30. Advances in Mathematics of Communications, 2009, 3 (4) : 311-316. doi: 10.3934/amc.2009.3.311

[12]

Katherine Morrison. An enumeration of the equivalence classes of self-dual matrix codes. Advances in Mathematics of Communications, 2015, 9 (4) : 415-436. doi: 10.3934/amc.2015.9.415

[13]

Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011

[14]

Suat Karadeniz, Bahattin Yildiz. New extremal binary self-dual codes of length $68$ from $R_2$-lifts of binary self-dual codes. Advances in Mathematics of Communications, 2013, 7 (2) : 219-229. doi: 10.3934/amc.2013.7.219

[15]

Steven T. Dougherty, Cristina Fernández-Córdoba. Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes. Advances in Mathematics of Communications, 2011, 5 (4) : 571-588. doi: 10.3934/amc.2011.5.571

[16]

Masaaki Harada. Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights. Advances in Mathematics of Communications, 2016, 10 (4) : 695-706. doi: 10.3934/amc.2016035

[17]

Annika Meyer. On dual extremal maximal self-orthogonal codes of Type I-IV. Advances in Mathematics of Communications, 2010, 4 (4) : 579-596. doi: 10.3934/amc.2010.4.579

[18]

Delphine Boucher. Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$. Advances in Mathematics of Communications, 2016, 10 (4) : 765-795. doi: 10.3934/amc.2016040

[19]

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

[20]

Lars Eirik Danielsen. Graph-based classification of self-dual additive codes over finite fields. Advances in Mathematics of Communications, 2009, 3 (4) : 329-348. doi: 10.3934/amc.2009.3.329

2017 Impact Factor: 0.564

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]