November  2011, 5(4): 681-686. doi: 10.3934/amc.2011.5.681

All binary linear codes of lengths up to 18 or redundancy up to 10 are normal

1. 

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, P.O.Box 323, 5000 Veliko Tarnovo, Bulgaria

Received  December 2010 Revised  June 2011 Published  November 2011

We show that all binary codes of lengths 16, 17 and 18, or redundancy 10, are normal. These results have applications in the construction of codes that attain $t[n,k]$, the smallest covering radius of any binary linear code.
Citation: Tsonka Baicheva. All binary linear codes of lengths up to 18 or redundancy up to 10 are normal. Advances in Mathematics of Communications, 2011, 5 (4) : 681-686. doi: 10.3934/amc.2011.5.681
References:
[1]

I. Bouyukliev, What is Q-extension?, Serdica J. Comput., 1 (2007), 115-130.

[2]

A. E. Brouwer, Bounds on the size of linear codes, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 295-461.

[3]

G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, "Covering Codes,'' Elsevier Science B. V., North-Holland, 1997.

[4]

G. D. Cohen, M. G. Karpovsky, H. F. Mattson, Jr. and J. R. Schatz, Covering radius - survey and recent results, IEEE Trans. Inform. Theory, 31 (1985), 738-740. doi: 10.1109/TIT.1985.1057043.

[5]

G. D. Cohen, S. N. Litsyn, A. C. Lobstein and H. F. Mattson, Jr., Covering radius 1985-1994, AAECC, 8 (1997), 173-239. doi: 10.1007/s002000050061.

[6]

G. D. Cohen, A. C. Lobstein and N. J. A. Sloane, Further results on the covering radius of codes, IEEE Trans. Inform. Theory, 32 (1986), 680-694. doi: 10.1109/TIT.1986.1057227.

[7]

R. L. Graham and N. J. A. Sloane, On the covering radius of codes, IEEE Trans. Inform. Theory, 31 (1985), 385-401. doi: 10.1109/TIT.1985.1057039.

[8]

X.-D. Hou, Binary linear quasi-perfect codes codes are normal, IEEE Trans. Inform. Theory, 37 (1991), 378-379. doi: 10.1109/18.75258.

[9]

H. Janwa and H. F. Mattson, Jr., Some upper bounds on the covering radii of linear codes over $F_q$ and their applications, Des. Codes Crypt., 18 (1999), 163-181. doi: 10.1023/A:1008397405457.

[10]

K. E. Kilby and N. J. A. Sloane, On the covering radius problem for codes: I Bounds on normalized covering radius, II Codes of low dimension; normal and abnormal codes, SIAM J. Algebraic Discrete Methods, 8 (1987), 604-627. doi: 10.1137/0608049.

[11]

H. F. Mattson, Jr., An improved upper bound on covering radius, in "Applied Algebra, Algorithmics and Error-Correcting Codes,'' Springer, Berlin, (1986), 90-106.

show all references

References:
[1]

I. Bouyukliev, What is Q-extension?, Serdica J. Comput., 1 (2007), 115-130.

[2]

A. E. Brouwer, Bounds on the size of linear codes, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 295-461.

[3]

G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, "Covering Codes,'' Elsevier Science B. V., North-Holland, 1997.

[4]

G. D. Cohen, M. G. Karpovsky, H. F. Mattson, Jr. and J. R. Schatz, Covering radius - survey and recent results, IEEE Trans. Inform. Theory, 31 (1985), 738-740. doi: 10.1109/TIT.1985.1057043.

[5]

G. D. Cohen, S. N. Litsyn, A. C. Lobstein and H. F. Mattson, Jr., Covering radius 1985-1994, AAECC, 8 (1997), 173-239. doi: 10.1007/s002000050061.

[6]

G. D. Cohen, A. C. Lobstein and N. J. A. Sloane, Further results on the covering radius of codes, IEEE Trans. Inform. Theory, 32 (1986), 680-694. doi: 10.1109/TIT.1986.1057227.

[7]

R. L. Graham and N. J. A. Sloane, On the covering radius of codes, IEEE Trans. Inform. Theory, 31 (1985), 385-401. doi: 10.1109/TIT.1985.1057039.

[8]

X.-D. Hou, Binary linear quasi-perfect codes codes are normal, IEEE Trans. Inform. Theory, 37 (1991), 378-379. doi: 10.1109/18.75258.

[9]

H. Janwa and H. F. Mattson, Jr., Some upper bounds on the covering radii of linear codes over $F_q$ and their applications, Des. Codes Crypt., 18 (1999), 163-181. doi: 10.1023/A:1008397405457.

[10]

K. E. Kilby and N. J. A. Sloane, On the covering radius problem for codes: I Bounds on normalized covering radius, II Codes of low dimension; normal and abnormal codes, SIAM J. Algebraic Discrete Methods, 8 (1987), 604-627. doi: 10.1137/0608049.

[11]

H. F. Mattson, Jr., An improved upper bound on covering radius, in "Applied Algebra, Algorithmics and Error-Correcting Codes,'' Springer, Berlin, (1986), 90-106.

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