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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

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.  doi: 10.1109/18.782168.  Google Scholar

[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.  doi: 10.1109/18.761269.  Google Scholar

[3]

M. C. Bhandari, M. K. Gupta and A. K. Lal, On $\mathbb Z_4$ simplex codes and their gray images,, in Applied Algebra, (1999), 170.  doi: 10.1007/3-540-46796-3_17.  Google Scholar

[4]

A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain, Type II codes over $\mathbb Z_4$,, IEEE Trans. Inform. Theory, 43 (1997), 969.  doi: 10.1109/18.568705.  Google Scholar

[5]

A. Bonnecaze, P. Solé and A. R. Calderbank, Quaternary quadratic residue codes and unimodular lattices,, IEEE Trans. Inform. Theory, 41 (1995), 366.  doi: 10.1109/18.370138.  Google Scholar

[6]

C. Carlet, $\mathbb Z_{2^k}$-linear codes,, IEEE Trans. Inform. Theory, 44 (1998), 1543.  doi: 10.1109/18.681328.  Google Scholar

[7]

C. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes,, Elsevier, (1997).   Google Scholar

[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.  doi: 10.1109/TIT.1985.1057043.  Google Scholar

[9]

C. J. Colbourn and M. K. Gupta, On quaternary MacDonald codes,, in Proc. Information Technology: Coding and Computing, (2003), 212.  doi: 10.1109/ITCC.2003.1197528.  Google Scholar

[10]

I. Constantinescu, W. Heise and T. Honold, Monomial extensions of isometries between codes over $\mathbb Z_m$,, in Proc. Workshop ACCT'96, (1996), 98.   Google Scholar

[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.  doi: 10.1016/0097-3165(93)90070-O.  Google Scholar

[12]

S. Dodunekov and J. Simonis, Codes and projective multisets,, Electr. J. Combin., 5 (1998).   Google Scholar

[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.  doi: 10.1023/A:1018696102510.  Google Scholar

[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.  doi: 10.14492/hokmj/1351001213.  Google Scholar

[15]

S. T. Dougherty, M. Harada and P. Solé, Shadow codes over $\mathbb Z_4$,, Finite Fields Appl., 7 (2001), 507.  doi: 10.1006/ffta.2000.0312.  Google Scholar

[16]

C. Durairajan, On Covering Codes and Covering Radius of Some Optimal Codes,, Ph.D Thesis, (1996).   Google Scholar

[17]

T. A. Gulliver and M. Harada, Double circulant self dual codes over $\mathbb Z_{2k}$,, IEEE Trans. Inform. Theory, 44 (1998), 3105.  doi: 10.1109/18.737540.  Google Scholar

[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.  doi: 10.1090/S0273-0979-1993-00426-9.  Google Scholar

[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.  doi: 10.1109/18.312154.  Google Scholar

[20]

M. Harada, New extremal Type II codes over $\mathbb Z_4$,, Des. Codes Cryptogr., 13 (1998), 271.  doi: 10.1023/A:1008254008212.  Google Scholar

[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), (1998).   Google Scholar

[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.  doi: 10.1109/18.556679.  Google Scholar

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.  doi: 10.1109/18.782168.  Google Scholar

[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.  doi: 10.1109/18.761269.  Google Scholar

[3]

M. C. Bhandari, M. K. Gupta and A. K. Lal, On $\mathbb Z_4$ simplex codes and their gray images,, in Applied Algebra, (1999), 170.  doi: 10.1007/3-540-46796-3_17.  Google Scholar

[4]

A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain, Type II codes over $\mathbb Z_4$,, IEEE Trans. Inform. Theory, 43 (1997), 969.  doi: 10.1109/18.568705.  Google Scholar

[5]

A. Bonnecaze, P. Solé and A. R. Calderbank, Quaternary quadratic residue codes and unimodular lattices,, IEEE Trans. Inform. Theory, 41 (1995), 366.  doi: 10.1109/18.370138.  Google Scholar

[6]

C. Carlet, $\mathbb Z_{2^k}$-linear codes,, IEEE Trans. Inform. Theory, 44 (1998), 1543.  doi: 10.1109/18.681328.  Google Scholar

[7]

C. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes,, Elsevier, (1997).   Google Scholar

[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.  doi: 10.1109/TIT.1985.1057043.  Google Scholar

[9]

C. J. Colbourn and M. K. Gupta, On quaternary MacDonald codes,, in Proc. Information Technology: Coding and Computing, (2003), 212.  doi: 10.1109/ITCC.2003.1197528.  Google Scholar

[10]

I. Constantinescu, W. Heise and T. Honold, Monomial extensions of isometries between codes over $\mathbb Z_m$,, in Proc. Workshop ACCT'96, (1996), 98.   Google Scholar

[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.  doi: 10.1016/0097-3165(93)90070-O.  Google Scholar

[12]

S. Dodunekov and J. Simonis, Codes and projective multisets,, Electr. J. Combin., 5 (1998).   Google Scholar

[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.  doi: 10.1023/A:1018696102510.  Google Scholar

[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.  doi: 10.14492/hokmj/1351001213.  Google Scholar

[15]

S. T. Dougherty, M. Harada and P. Solé, Shadow codes over $\mathbb Z_4$,, Finite Fields Appl., 7 (2001), 507.  doi: 10.1006/ffta.2000.0312.  Google Scholar

[16]

C. Durairajan, On Covering Codes and Covering Radius of Some Optimal Codes,, Ph.D Thesis, (1996).   Google Scholar

[17]

T. A. Gulliver and M. Harada, Double circulant self dual codes over $\mathbb Z_{2k}$,, IEEE Trans. Inform. Theory, 44 (1998), 3105.  doi: 10.1109/18.737540.  Google Scholar

[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.  doi: 10.1090/S0273-0979-1993-00426-9.  Google Scholar

[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.  doi: 10.1109/18.312154.  Google Scholar

[20]

M. Harada, New extremal Type II codes over $\mathbb Z_4$,, Des. Codes Cryptogr., 13 (1998), 271.  doi: 10.1023/A:1008254008212.  Google Scholar

[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), (1998).   Google Scholar

[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.  doi: 10.1109/18.556679.  Google Scholar

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