# American Institute of Mathematical Sciences

2012, 6(4): 499-503. doi: 10.3934/amc.2012.6.499

## Existence of cyclic self-orthogonal codes: A note on a result of Vera Pless

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

Received  January 2012 Revised  June 2012 Published  November 2012

It is of interest to know when cyclic self-orthogonal codes of length $n$ over $\mathbb F_q$ do not exist. The conditions, listed by Pless in [7] under which cyclic self-orthogonal codes can not exist, are not always sufficient. An example is given to assert this. Here we give the necessary and sufficient conditions under which cyclic self-orthogonal codes of length $n$ over $\mathbb F_q$ do not exist.
Citation: Leetika Kathuria, Madhu Raka. Existence of cyclic self-orthogonal codes: A note on a result of Vera Pless. Advances in Mathematics of Communications, 2012, 6 (4) : 499-503. doi: 10.3934/amc.2012.6.499
##### 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., (). doi: 10.1016/j.ffa.2012.10.003. [3] D. M. Burton, "Elementry Number Theory,'' 6th edition,, Tata McGraw-Hill, (2006). [4] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'', Cambridge, (2003). doi: 10.1017/CBO9780511807077. [5] 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. [6] X. S. Kai and S. X. Zhu, On cyclic self-dual codes,, Appl. Algebra Engrg. Comm. Comput., 19 (2008), 509. doi: 10.1007/s00200-008-0086-9. [7] V. Pless, Cyclotomy and cyclic codes, the unreasonable effectiveness of number theory,, in, 46 (1992), 91. [8] E. M. Rains and N. J. A. Sloane, Self-dual codes,, in, (1998), 177. [9] N. J. A. Sloane and J. G. Thompson, Cyclic self-dual codes,, IEEE Trans. Inform. Theory, 29 (1983), 364. doi: 10.1109/TIT.1983.1056682.

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., (). doi: 10.1016/j.ffa.2012.10.003. [3] D. M. Burton, "Elementry Number Theory,'' 6th edition,, Tata McGraw-Hill, (2006). [4] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'', Cambridge, (2003). doi: 10.1017/CBO9780511807077. [5] 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. [6] X. S. Kai and S. X. Zhu, On cyclic self-dual codes,, Appl. Algebra Engrg. Comm. Comput., 19 (2008), 509. doi: 10.1007/s00200-008-0086-9. [7] V. Pless, Cyclotomy and cyclic codes, the unreasonable effectiveness of number theory,, in, 46 (1992), 91. [8] E. M. Rains and N. J. A. Sloane, Self-dual codes,, in, (1998), 177. [9] N. J. A. Sloane and J. G. Thompson, Cyclic self-dual codes,, IEEE Trans. Inform. Theory, 29 (1983), 364. doi: 10.1109/TIT.1983.1056682.
 [1] 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 [2] 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 [3] Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031 [4] 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 [5] 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 [6] 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 [7] 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 [8] Dean Crnković, Bernardo Gabriel Rodrigues, Sanja Rukavina, Loredana Simčić. Self-orthogonal codes from orbit matrices of 2-designs. Advances in Mathematics of Communications, 2013, 7 (2) : 161-174. doi: 10.3934/amc.2013.7.161 [9] Crnković Dean, Vedrana Mikulić Crnković, Bernardo G. Rodrigues. On self-orthogonal designs and codes related to Held's simple group. Advances in Mathematics of Communications, 2018, 12 (3) : 607-628. doi: 10.3934/amc.2018036 [10] 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 [11] Xia Li, Feng Cheng, Chunming Tang, Zhengchun Zhou. Some classes of LCD codes and self-orthogonal codes over finite fields. Advances in Mathematics of Communications, 2019, 13 (2) : 267-280. doi: 10.3934/amc.2019018 [12] W. Cary Huffman. Self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes with an automorphism of prime order. Advances in Mathematics of Communications, 2013, 7 (1) : 57-90. doi: 10.3934/amc.2013.7.57 [13] W. Cary Huffman. Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order. Advances in Mathematics of Communications, 2007, 1 (3) : 357-398. doi: 10.3934/amc.2007.1.357 [14] Nikolay Yankov. Self-dual [62, 31, 12] and [64, 32, 12] codes with an automorphism of order 7. Advances in Mathematics of Communications, 2014, 8 (1) : 73-81. doi: 10.3934/amc.2014.8.73 [15] 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 [16] 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 [17] 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 [18] 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 [19] 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 [20] 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

2017 Impact Factor: 0.564