Type I and type II codes over the ring \(\mathbb{F}_2+s \mathbb{F}_2+s^2\mathbb{F}_2\). (English) Zbl 1456.94130
Summary: We take a ring \(R = \mathbb{F}_2+s\mathbb{F}_2+s^2\mathbb{F}_2\). We consider a Gray map on this ring, discuss self-dual codes, define various weight enumerators over the ring, and discuss equivalence class of codes over the ring. We construct self-dual codes of Type I and Type II over the given ring for different lengths.
MSC:
| 94B05 | Linear codes (general theory) |
References:
| [1] | Bannai, E.; Harada, M.; Ibukiyama, T.; Munemasa, A.; Oura, M.: Type II codes over \({{\mathbb{F}}_2} +u{{\mathbb{F}}_2}\) and applications to Hermitian modular forms, Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg. 73, Springer, 13-42 (2003) · Zbl 1048.11038 |
| [2] | Bonnecaze, A.; Udaya, P., Cyclic codes and self-dual codes over \({\mathbb{F}}_2 + u{\mathbb{F}}_2\), IEEE Trans. Inf. Theory, 45, 4, 1250-1255 (1999) · Zbl 0958.94025 · doi:10.1109/18.761278 |
| [3] | Choie, Y.; Kim, N., The complete weight enumerator of Type II code over \({\mathbb{Q}}(\sqrt{5})\) and Jacobi forms, IEEE Trans. Inf. Theory, 47, 1, 397 (2001) |
| [4] | Conway, Jh; Pless, V.; Sloane, Nja, The binary self-dual codes of length up to \(32:\) a revised enumeration, J. Comb. Theory Ser. A., 60, 2, 183-195 (1992) · Zbl 0751.94009 · doi:10.1016/0097-3165(92)90003-D |
| [5] | Dougherty, S.T.: Algebraic Coding Theory Over Finite Commutative Rings. Springer (2017) · Zbl 1375.94002 |
| [6] | Dougherty, St; Gaborit, P.; Harada, M.; Sole, P., Type II codes over \({\mathbb{F}}_2+u{\mathbb{F}}_2\), IEEE Trans. Inf. Theory, 45, 1, 32-45 (1999) · Zbl 0947.94023 · doi:10.1109/18.746770 |
| [7] | Fields, J.; Pless, V.; Leon, J. S.: All self-dual \({{\mathbb{Z}}_4}\) codes of length \(15\) or less are known. In: IEEE International Symposium on Information Theory, p 202 (1997) · Zbl 0899.94027 |
| [8] | Gaborit, P., Mass formulas for self-dual codes over \({\mathbb{Z}}_4\) and \({\mathbb{F}}_q +u{\mathbb{F}}_q\) rings, IEEE Trans. Inf. Theory, 42, 4, 122-128 (1996) · Zbl 0854.94013 · doi:10.1109/18.508845 |
| [9] | Huffman, Wc, On the decomposition of self-dual codes over \({\mathbb{F}}_2 + u{\mathbb{F}}_2\) with an automorphism of odd prime order, Finite Fields Their Appl., 13, 3, 681-712 (2007) · Zbl 1120.94011 · doi:10.1016/j.ffa.2006.02.003 |
| [10] | Kewat, Pk; Ghosh, B.; Pattanayak, S., Cyclic codes over the ring \({\mathbb{Z}}_p[u, v]/ < u^2, v^2, uv- vu>\), Finite Fields Their Appl., 34, 161-175 (2015) · Zbl 1392.94941 · doi:10.1016/j.ffa.2015.01.005 |
| [11] | Kim, Hj; Lee, Y., Construction of extremal self-dual codes over \({\mathbb{F}}_2 + u{\mathbb{F}}_2\) with an automorphism of odd order, Finite Fields and Their Appl., 18, 5, 971-992 (2012) · Zbl 1270.94068 · doi:10.1016/j.ffa.2012.05.004 |
| [12] | MacWilliams, F.J.; Sloane, N.J.A.: The Theory of Error Correcting Codes. Elsevier (1977) · Zbl 0369.94008 |
| [13] | Mallows, Cl; Sloane, N., An upper bound for self-dual codes, Inf. Control, 22, 2, 188-200 (1973) · Zbl 0254.94011 · doi:10.1016/S0019-9958(73)90273-8 |
| [14] | Pless, V.; Brualdi, R.A.; Huffman, W.C.: Handbook of coding theory. Elsevier Science Inc. (1998) · Zbl 0907.94001 |
| [15] | Pless, V.; Huffman, W.C.: Fundamentals of error-correcting codes. Cambridge University Press (2003) · Zbl 1099.94030 |
| [16] | Pless, V.; Leon, Js; Fields, J., All \({\mathbb{Z}}_4\) codes of Type II and length \(16\) are known, J. Comb. Theory Ser. A., 78, 1, 32-50 (1997) · Zbl 0872.94050 · doi:10.1006/jcta.1996.2750 |
| [17] | Qian, J.; Zhang, L.; Yin, Z.: Type II code over \({\mathbb{F}}_2 + u{\mathbb{F}}_2 + u^2{\mathbb{F}}_2\). In: Information Theory Workshop, IEEE. pp 21-23 (2006) |
| [18] | Singh, Ak; Kewat, Pk, On cyclic codes over the ring \({\mathbb{Z}}_p[u]/ < u^k >\), Des. Codes Cryptogr., 74, 1, 1-13 (2015) · Zbl 1351.94089 · doi:10.1007/s10623-013-9843-2 |
| [19] | Wood, Ja, Duality for modules over finite rings and applications to coding theory, Am. J. Math., 121, 3, 555-575 (1999) · Zbl 0968.94012 · doi:10.1353/ajm.1999.0024 |
| [20] | Yildiz, B.; Karadeniz, S., Linear codes over \({\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2\), Des. Codes Cryptogr., 54, 1, 61-81 (2010) · Zbl 1187.94045 · doi:10.1007/s10623-009-9309-8 |
| [21] | Yildiz, B.; Karadeniz, S., Self-dual codes over \({\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2\), J. Franklin Inst., 347, 10, 1888-1894 (2010) · Zbl 1204.94103 · doi:10.1016/j.jfranklin.2010.10.007 |
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