COVERING RADIUS OF THE REPETITION CODES OVER Z2Z2^s
Subject Areas : Statistics
Fariba Mahmoudi
1
,
LOTFALLAH POURFARAJ
2
1 - Department of Mathematics, Central Tehran Branch Islamic Azad University, , Tehran. Iran.
2 - Department of Mathematics, Central Tehran BranchIslamic Azad University,,Tehran, Iran
Keywords: , ", شعاع پوششی", Z2Z2^s- کدهای جمعی", ", کدهای تکرار, نگاشت گری",
Abstract :
The covering radius of code C is the smallest number r such that the spheres of radius r around the codewords cover space. For a binary code C the covering radius r(C) is defined as follows: r(C) = maxu∈Z2{minc∈CdH(u, c)}. The extension of this definition to codes over Z2Z2^s, is that, the covering radius of a code C is the smallest number r such that the spheres of radius r around the codewords cover (Z2Z2^s)^n. Then the covering radius of a code cover Z2Z2s, with respect to the Lee distances, is given by rL(C) = maxu∈Z2 ×Z2 ^s{minc∈CdL(u, c)}, The covering radius is important for determining the error correcting capability of these codes. We determine the exact covering radius of the various repetition codes, which have been constructed using the zero divisors and units in Z2Z2^s . .Also, we determine the exact covering radius of the various repetition codes over Z2Z2^s .
[1] P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Reports Supplements 10, (1973).
[2] P. Delsarte, V. Levenshtein, Association schemes and coding theory, IEEE Transaction on Information Theory 44 (6): 2477–2504(1998).
[3] J. Pujol, J. Rifa, Translation invariant propelinear codes, IEEE Transaction on Information Theory 10: 116–118 (1964).
[4] I. Aydogdu and I. Siap, The structure of -additive codes: Bounds on the minimum distance, Applied Mathematics and Information Sciences (AMIS) 7: 2271–2278) 2013).
[5] C. Cohenand I. Honkala, S. Litsyn and A. Lobstein, Covering radius, Elsevier, (1997).
[6] T. Aoki, P. Gaborit, M. Harada, M. Ozeki, P. Solé, On the covering radius of -codes and their lattices, IEEE Transaction Information Theory 45 (6): 2162–2168(1999).
[7] K. Chatouh, K. Guenda, T. Aaron Gulliver, L. Noui, On some classes of linear codes over and their covering radii, Journal of Applied Mathematics and Computing 53: 201-222 (2017).
[8]M. Curz,C. Durairajan and P. Sole, On the covering radius of codes over , Mathematics, MDPI 2020 8 (3): pp.328(2020)
[9] M. Bilal, J. Borges, S.T. Dougherty, C. Fernández-Córdoba, Maximum distance separable codes over and , Designs, Codes and Cryptography 61 (1): 31–40(2011).
[10] J. Borges, Fernández-Córdoba, C. Pujol, J. Rifá, J. Villanueva, linear codes: Generator matrices and duality. Designs, Codes and Cryptography 54, (2): 167-179(2010).
[11] S. T. Dougherty and C. Fernandez-C. ordoba, Codes over gray map and self-dual codes, Advances in Mathematics of Communications 5: 571-588(2011).
[12] M.K. Gupta, C. Durairajan, On the Covering Radius of Some moudular codes, arXiv:1206.3038v2 [cs.IT ](2012).
[13] Durairajan, C. On Covering Codes and Covering Radius of Some Optimal codes. Ph.D. Thesis, Department of Mathematics, IIT Kanpur, Kanpur, India, (1996).
