On the covering radius of some modular codes

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May 2014, 8(2): 129-137. doi: 10.3934/amc.2014.8.129

On the covering radius of some modular codes

Manish K. Gupta ^1, and Chinnappillai Durairajan ^2,

1.	Laboratory of Natural Information Processing, Dhirubhai Ambani Institute of Information and Communication Technology, Gandhinagar, Gujarat 382007, India
2.	Department of Mathematics, School of Mathematical Sciences, Bharathidasan University, Tiruchirappalli, Tamil Nadu 620024, India

Received July 2012 Published May 2014

Abstract
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This paper gives lower and upper bounds on the covering radius of codes over $\mathbb{Z}_{2^s}$ with respect to homogenous distance. We also determine the covering radius of various Repetition codes, Simplex codes (Type $\alpha$ and Type $\beta$) and their dual and give bounds on the covering radii for MacDonald codes of both types over $\mathbb{Z}_4$.

Keywords: simplex codes, Covering radius, codes over rings, Hamming codes..

Mathematics Subject Classification: Primary: 94B25; Secondary: 11H3.

Citation: Manish K. Gupta, Chinnappillai Durairajan. On the covering radius of some modular codes. Advances in Mathematics of Communications, 2014, 8 (2) : 129-137. doi: 10.3934/amc.2014.8.129

References:

[1]	T. Aoki, P. Gaborit, M. Harada, M. Ozeki and P. Solé, On the covering radius of $\mathbb Z_{4}$ codes and their lattices, IEEE Trans. Inform. Theory, 45 (1999), 2162-2168. doi: 10.1109/18.782168.
[2]	E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type II codes, even unimodular lattices and invariant rings, IEEE Trans. Inform. Theory, 45 (1999), 1194-1205. doi: 10.1109/18.761269.
[3]	M. C. Bhandari, M. K. Gupta and A. K. Lal, On $\mathbb Z_4$ simplex codes and their gray images, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer, 1999, 170-179. doi: 10.1007/3-540-46796-3_17.
[4]	A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain, Type II codes over $\mathbb Z_4$, IEEE Trans. Inform. Theory, 43 (1997), 969-976. doi: 10.1109/18.568705.
[5]	A. Bonnecaze, P. Solé and A. R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory, 41 (1995), 366-377. doi: 10.1109/18.370138.
[6]	C. Carlet, $\mathbb Z_{2^k}$-linear codes, IEEE Trans. Inform. Theory, 44 (1998), 1543-1547. doi: 10.1109/18.681328.
[7]	C. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, Elsevier, 1997.
[8]	G. D. Cohen, M. G. Karpovsky, H. F. Mattson and J. R. Schatz, Covering radius-survey and recent results, IEEE Trans. Inform. Theory, 31 (1985), 328-343. doi: 10.1109/TIT.1985.1057043.
[9]	C. J. Colbourn and M. K. Gupta, On quaternary MacDonald codes, in Proc. Information Technology: Coding and Computing, 2003, 212-215. doi: 10.1109/ITCC.2003.1197528.
[10]	I. Constantinescu, W. Heise and T. Honold, Monomial extensions of isometries between codes over $\mathbb Z_m$, in Proc. Workshop ACCT'96, Sozopol, Bulgaria, 1996, 98-104.
[11]	J. H. Conway and N. J. A. Sloane, Self-dual codes over the integers modulo $4$, J. Combin. Theory Ser. A, 62 (1993), 30-45. doi: 10.1016/0097-3165(93)90070-O.
[12]	S. Dodunekov and J. Simonis, Codes and projective multisets, Electr. J. Combin., 5 (1998), R37.
[13]	S. T. Dougherty, T. A. Gulliver and M. Harada, Type II codes over finite rings and even unimodular lattices, J. Alg. Combin., 9 (1999), 233-250. doi: 10.1023/A:1018696102510.
[14]	S. T. Dougherty, M. Harada and P. Solé, Self-dual codes over rings and the Chinese remainder theorem, Hokkaido Math. J., 28 (1999), 253-283. doi: 10.14492/hokmj/1351001213.
[15]	S. T. Dougherty, M. Harada and P. Solé, Shadow codes over $\mathbb Z_4$, Finite Fields Appl., 7 (2001), 507-529. doi: 10.1006/ffta.2000.0312.
[16]	C. Durairajan, On Covering Codes and Covering Radius of Some Optimal Codes, Ph.D Thesis, Department of Mathematics, IIT Kanpur, 1996.
[17]	T. A. Gulliver and M. Harada, Double circulant self dual codes over $\mathbb Z_{2k}$, IEEE Trans. Inform. Theory, 44 (1998), 3105-3123. doi: 10.1109/18.737540.
[18]	A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, A linear construction for certain Kerdock and Preparata codes, Bull Amer. Math. Soc., 29 (1993), 218-222. doi: 10.1090/S0273-0979-1993-00426-9.
[19]	A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of kerdock, preparata, goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.
[20]	M. Harada, New extremal Type II codes over $\mathbb Z_4$, Des. Codes Cryptogr., 13 (1998), 271-284. doi: 10.1023/A:1008254008212.
[21]	E. M. Rains and N. J. A. Sloane, Self-dual codes, in The Handbook of Coding Theory (eds. V. Pless and W.C. Huffman), North-Holland, 1998.
[22]	V. V. Vazirani, H. Saran and B. SundarRajan, An efficient algorithm for constructing minimal trellises for codes over finite abelian groups, IEEE Trans. Inform. Theory, 42 (1996), 1839-1854. doi: 10.1109/18.556679.

show all references

References:

[1]	T. Aoki, P. Gaborit, M. Harada, M. Ozeki and P. Solé, On the covering radius of $\mathbb Z_{4}$ codes and their lattices, IEEE Trans. Inform. Theory, 45 (1999), 2162-2168. doi: 10.1109/18.782168.
[2]	E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type II codes, even unimodular lattices and invariant rings, IEEE Trans. Inform. Theory, 45 (1999), 1194-1205. doi: 10.1109/18.761269.
[3]	M. C. Bhandari, M. K. Gupta and A. K. Lal, On $\mathbb Z_4$ simplex codes and their gray images, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer, 1999, 170-179. doi: 10.1007/3-540-46796-3_17.
[4]	A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain, Type II codes over $\mathbb Z_4$, IEEE Trans. Inform. Theory, 43 (1997), 969-976. doi: 10.1109/18.568705.
[5]	A. Bonnecaze, P. Solé and A. R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory, 41 (1995), 366-377. doi: 10.1109/18.370138.
[6]	C. Carlet, $\mathbb Z_{2^k}$-linear codes, IEEE Trans. Inform. Theory, 44 (1998), 1543-1547. doi: 10.1109/18.681328.
[7]	C. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, Elsevier, 1997.
[8]	G. D. Cohen, M. G. Karpovsky, H. F. Mattson and J. R. Schatz, Covering radius-survey and recent results, IEEE Trans. Inform. Theory, 31 (1985), 328-343. doi: 10.1109/TIT.1985.1057043.
[9]	C. J. Colbourn and M. K. Gupta, On quaternary MacDonald codes, in Proc. Information Technology: Coding and Computing, 2003, 212-215. doi: 10.1109/ITCC.2003.1197528.
[10]	I. Constantinescu, W. Heise and T. Honold, Monomial extensions of isometries between codes over $\mathbb Z_m$, in Proc. Workshop ACCT'96, Sozopol, Bulgaria, 1996, 98-104.
[11]	J. H. Conway and N. J. A. Sloane, Self-dual codes over the integers modulo $4$, J. Combin. Theory Ser. A, 62 (1993), 30-45. doi: 10.1016/0097-3165(93)90070-O.
[12]	S. Dodunekov and J. Simonis, Codes and projective multisets, Electr. J. Combin., 5 (1998), R37.
[13]	S. T. Dougherty, T. A. Gulliver and M. Harada, Type II codes over finite rings and even unimodular lattices, J. Alg. Combin., 9 (1999), 233-250. doi: 10.1023/A:1018696102510.
[14]	S. T. Dougherty, M. Harada and P. Solé, Self-dual codes over rings and the Chinese remainder theorem, Hokkaido Math. J., 28 (1999), 253-283. doi: 10.14492/hokmj/1351001213.
[15]	S. T. Dougherty, M. Harada and P. Solé, Shadow codes over $\mathbb Z_4$, Finite Fields Appl., 7 (2001), 507-529. doi: 10.1006/ffta.2000.0312.
[16]	C. Durairajan, On Covering Codes and Covering Radius of Some Optimal Codes, Ph.D Thesis, Department of Mathematics, IIT Kanpur, 1996.
[17]	T. A. Gulliver and M. Harada, Double circulant self dual codes over $\mathbb Z_{2k}$, IEEE Trans. Inform. Theory, 44 (1998), 3105-3123. doi: 10.1109/18.737540.
[18]	A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, A linear construction for certain Kerdock and Preparata codes, Bull Amer. Math. Soc., 29 (1993), 218-222. doi: 10.1090/S0273-0979-1993-00426-9.
[19]	A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of kerdock, preparata, goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.
[20]	M. Harada, New extremal Type II codes over $\mathbb Z_4$, Des. Codes Cryptogr., 13 (1998), 271-284. doi: 10.1023/A:1008254008212.
[21]	E. M. Rains and N. J. A. Sloane, Self-dual codes, in The Handbook of Coding Theory (eds. V. Pless and W.C. Huffman), North-Holland, 1998.
[22]	V. V. Vazirani, H. Saran and B. SundarRajan, An efficient algorithm for constructing minimal trellises for codes over finite abelian groups, IEEE Trans. Inform. Theory, 42 (1996), 1839-1854. doi: 10.1109/18.556679.

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