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Volume 22, No 1, 2015, P. 32-50

UDC 519.8
S. A. Malyugin
Affine 3-nonsystematic perfect codes of length 15

Abstract:
A perfect binary code C of length n = 2k − 1 is called affine 3-systematic if there exists a 3-dimensional subspace L in the space {0, 1}n such that the intersection of any of its cosets L + u with C is either empty, or a singleton. Otherwise, the code C is called affine 3-nonsystematic. In the paper, we construct four nonequivalent affine 3-nonsystematic codes of length 15.
Bibliogr. 12.

Keywords: perfect code, Hamming code, nonsystematic code, affine nonsystematic code, affine 3-nonsystematic code, component.

DOI: 10.17377/daio.2015.22.438

Sergey A. Malyugin1
1. Sobolev Institute of Mathematics
4 Koptyug Ave., 630090 Novosibirsk, Russia
ň-mail: mal@math.nsc.ru

Received 26 January 2014
Revised 24 September 2014

References

[1] S. V. Avgustinovich and F. I. Solov’eva, On nonsystematic perfect binary codes, Probl. Peredachi Inf., 32, No. 3, 47–50, 1996. Translated in Probl. Inf. Transm., 32, No. 3, 47–50, 1996.

[2] S. A. Malyugin, On enumeration of the perfect binary codes of length 15, Diskretn. Anal. Issled. Oper., Ser. 2, 6, No. 2, 48–73, 1999. Translated in Diskrete Appl. Math., 135, No. 1–3, 161–181, 2004.

[3] S. A. Malyugin, Nonsystematic perfect binary codes, Diskretn. Anal. Issled. Oper., Ser. 1, 8, No. 1, 55–76, 2001.

[4] S. A. Malyugin, On affine nonsystematic codes, Proc. Conf. Devoted to the 90th Anniversary of A.A. Lyapunov, Novosibirsk, Russia, Oct. 8–11, 2001, pp. 393–394, Novosibirsk, 2009. Available at
http://www.sbras.ru/ws/Lyap2001/2288/. Accessed Jan. 13, 2015.

[5] S. A. Malyugin, On enumeration of nonequivalent perfect binary codes of length 15 and rank 15, Diskretn. Anal. Issled. Oper., Ser. 1, 13, No. 1, 77–98, 2006. Translated in J. Appl. Ind. Math.,1, No. 1, 77–80, 2007.

[6] S. A. Malyugin, Affine nonsystematic codes, Diskretn. Anal. Issled. Oper., 19, No. 4, 73–85, 2012. Translated in J. Appl. Ind. Math.,6, No. 4, 451–459, 2012.

[7] S. A. Malyugin, Affine 3-nonsystematic codes, Diskretn. Anal. Issled. Oper., 21, No. 4, 54–61, 2014. Translated in J. Appl. Ind. Math., 8, No. 4, 552–556, 2014.

[8] A. M. Romanov, On construction of perfect nonlinear binary codes by symbol inversion, Diskretn. Anal. Issled. Oper., Ser. 1, 4, No. 1, 46–52, 1997.

[9.] A. M. Romanov, On nonsystematic perfect codes of length 15, Diskretn. Anal. Issled. Oper., Ser. 1, 4, No. 4, 75–78, 1997. Translated in Discrete Appl. Math., 135, No. 1–3, 255–258, 2004.

[10] K. T. Phelps and M. LeVan, Kernels of nonlinear Hamming codes, Des. Codes Cryptogr., 6, No. 3, 247–257, 1995.

[11] K. T. Phelps and M. LeVan, Nonsystematic perfect codes, SIAM J. Discrete Math., 12, No.1, 27–34, 1999.

[12] F. I. Solov’eva, Switchings and perfect codes, in I. AlthĘofer et al., eds., Numbers, Information and Complexity, pp. 311–324, Kluwer Acad. Publ., Dordrecht, 2000.
 © Sobolev Institute of Mathematics, 2015