On the covering radius of some modular codes

Previous Article
Nearest-neighbor entropy estimators with weak metrics
AMC Home
This Issue
Next Article
On Abelian group representability of finite groups

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
Related Papers

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.
[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.
[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.
[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.
[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.
[6]	C. Carlet, $\mathbb Z_{2^k}$-linear codes,, IEEE Trans. Inform. Theory, 44 (1998), 1543. 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. 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. 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, (1996), 98.
[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.
[12]	S. Dodunekov and J. Simonis, Codes and projective multisets,, Electr. J. Combin., 5 (1998).
[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.
[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.
[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.
[16]	C. Durairajan, On Covering Codes and Covering Radius of Some Optimal Codes,, Ph.D Thesis, (1996).
[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.
[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.
[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.
[20]	M. Harada, New extremal Type II codes over $\mathbb Z_4$,, Des. Codes Cryptogr., 13 (1998), 271. 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)*, (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. 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. 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. 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, (1999), 170. 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. 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. doi: 10.1109/18.370138.
[6]	C. Carlet, $\mathbb Z_{2^k}$-linear codes,, IEEE Trans. Inform. Theory, 44 (1998), 1543. 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. 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. 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, (1996), 98.
[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.
[12]	S. Dodunekov and J. Simonis, Codes and projective multisets,, Electr. J. Combin., 5 (1998).
[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.
[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.
[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.
[16]	C. Durairajan, On Covering Codes and Covering Radius of Some Optimal Codes,, Ph.D Thesis, (1996).
[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.
[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.
[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.
[20]	M. Harada, New extremal Type II codes over $\mathbb Z_4$,, Des. Codes Cryptogr., 13 (1998), 271. 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)*, (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. doi: 10.1109/18.556679.

[1]	Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409
[2]	Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025
[3]	Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004
[4]	Tsonka Baicheva, Iliya Bouyukliev. On the least covering radius of binary linear codes of dimension 6. Advances in Mathematics of Communications, 2010, 4 (3) : 399-404. doi: 10.3934/amc.2010.4.399
[5]	Aicha Batoul, Kenza Guenda, T. Aaron Gulliver. Some constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2016, 10 (4) : 683-694. doi: 10.3934/amc.2016034
[6]	Somphong Jitman, San Ling, Patanee Udomkavanich. Skew constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2012, 6 (1) : 39-63. doi: 10.3934/amc.2012.6.39
[7]	Kanat Abdukhalikov. On codes over rings invariant under affine groups. Advances in Mathematics of Communications, 2013, 7 (3) : 253-265. doi: 10.3934/amc.2013.7.253
[8]	Delphine Boucher, Patrick Solé, Felix Ulmer. Skew constacyclic codes over Galois rings. Advances in Mathematics of Communications, 2008, 2 (3) : 273-292. doi: 10.3934/amc.2008.2.273
[9]	Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395
[10]	Steven T. Dougherty, Esengül Saltürk, Steve Szabo. Codes over local rings of order 16 and binary codes. Advances in Mathematics of Communications, 2016, 10 (2) : 379-391. doi: 10.3934/amc.2016012
[11]	Alonso sepúlveda Castellanos. Generalized Hamming weights of codes over the $\mathcal{GH}$ curve. Advances in Mathematics of Communications, 2017, 11 (1) : 115-122. doi: 10.3934/amc.2017006
[12]	Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Addendum. Advances in Mathematics of Communications, 2011, 5 (3) : 543-546. doi: 10.3934/amc.2011.5.543
[13]	Joaquim Borges, Josep Rifà, Victor Zinoviev. Completely regular codes by concatenating Hamming codes. Advances in Mathematics of Communications, 2018, 12 (2) : 337-349. doi: 10.3934/amc.2018021
[14]	Thomas Westerbäck. Parity check systems of nonlinear codes over finite commutative Frobenius rings. Advances in Mathematics of Communications, 2017, 11 (3) : 409-427. doi: 10.3934/amc.2017035
[15]	Steven T. Dougherty, Abidin Kaya, Esengül Saltürk. Cyclic codes over local Frobenius rings of order 16. Advances in Mathematics of Communications, 2017, 11 (1) : 99-114. doi: 10.3934/amc.2017005
[16]	Hai Q. Dinh, Hien D. T. Nguyen. On some classes of constacyclic codes over polynomial residue rings. Advances in Mathematics of Communications, 2012, 6 (2) : 175-191. doi: 10.3934/amc.2012.6.175
[17]	Zihui Liu, Dajian Liao. Higher weights and near-MDR codes over chain rings. Advances in Mathematics of Communications, 2018, 12 (4) : 761-772. doi: 10.3934/amc.2018045
[18]	Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011
[19]	Washiela Fish, Jennifer D. Key, Eric Mwambene. Partial permutation decoding for simplex codes. Advances in Mathematics of Communications, 2012, 6 (4) : 505-516. doi: 10.3934/amc.2012.6.505
[20]	David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131

2018 Impact Factor: 0.879

American Institute of Mathematical Sciences