American Institute of Mathematical Sciences

August  2017, 11(3): 409-427. doi: 10.3934/amc.2017035

Parity check systems of nonlinear codes over finite commutative Frobenius rings

 Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076 Aalto, Finland

Received  March 2012 Revised  February 2016 Published  August 2017

Fund Project: The author acknowledges support from the Knut and Alice Wallenberg Foundation under grant KAW 2005.0098. This support was given during the beginning of the work on this paper while the author was affiliated at the Department of Mathematics, KTH, S-10044 Stockholm, Sweden.

The concept of parity check matrices of linear binary codes has been extended by Heden [10] to parity check systems of nonlinear binary codes. In the present paper we extend this concept to parity check systems of nonlinear codes over finite commutative Frobenius rings. Using parity check systems, results on how to get some fundamental properties of the codes are given. Moreover, parity check systems and its connection to characters is investigated and a MacWilliams type theorem on the distance distribution is given.

Citation: 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
References:
 [1] T. Britz, MacWilliams identities and matroid polynomials, Electr. J. Combin., 9 (2002), R19, 16pp. [2] P. Delsarte, Bounds for unrestricted codes, by linear programming, Philips Res. Rep., 27 (1972), 272-289. [3] T. Etzion and A. Vardy, On perfect codes and tilings, problems and solutions, SIAM J. Discr. Math., 11 (1998), 205-223.  doi: 10.1137/S0895480196309171. [4] M. Greferath, An introduction to ring-linear coding theory, in Gröbner Bases, Coding and Cryptography (eds. M. Sala et al), Springer-Verlag, Berlin, 2009,219–238. [5] M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes, J. Alg. Appl., 3 (2004), 247-272.  doi: 10.1142/S0219498804000873. [6] M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams' equivalence theorem, J. Combin. Theory Ser. A, 92 (2000), 17-28.  doi: 10.1006/jcta.1999.3033. [7] A. R. Hammons, Jr., 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. Inf. Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154. [8] O. Heden, A full rank perfect code of length 31, Des. Codes Crypt., 38 (2006), 125-129.  doi: 10.1007/s10623-005-5665-1. [9] O. Heden, On perfect $p$-ary codes of length p+1, Des. Codes Crypt., 46 (2008), 45-56.  doi: 10.1007/s10623-007-9133-y. [10] O. Heden, Perfect codes from the dual point of view Ⅰ, Discr. Math., 308 (2008), 6141-6156.  doi: 10.1016/j.disc.2007.11.037. [11] M. Hessler, Perfect codes as isomorphic spaces, Discr. Math., 306 (2006), 1981-1987.  doi: 10.1016/j.disc.2006.03.039. [12] T. Honold, Characterization of finite Frobenius rings, Arch. Math., 76 (2001), 406-415.  doi: 10.1007/PL00000451. [13] T. Honold and A. A. Nechaev, Weighted modules and linear representations of codes, Probl. Inf. Transm., 35 (1999), 205-223. [14] T. Honold and I. Landjev, MacWilliams identities for linear codes over finite Frobenius rings, in Finite Fields and Applications (eds. D. Jungnickel et al), Springer-Verlag, Berlin, 2001, 276–292. [15] F. J. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell Sys. Tech. J., 42 (1963), 79-94.  doi: 10.1002/j.1538-7305.1963.tb04003.x. [16] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes North-Holland, Amsterdam, 1977. [17] A. A. Nechaev, Finite principal ideal rings, Mat. Sbornik, 20 (1973), 364-382. [18] A. A. Nechaev, Kerdock code in a cyclic form, Discr. Math. Appl., 1 (1991), 365-384.  doi: 10.1515/dma.1991.1.4.365. [19] R. Y. Sharp, Steps in Commutative Algebra 2nd edition, Cambridge Univ. Press, Cambridge, 2000. doi: 10.1017/CBO9780511626265. [20] A. Terras, Fourier Analysis on Finite Groups and Applications Cambridge Univ. Press, Cambridge, 1999. doi: 10.1017/CBO9780511626265. [21] M. Villanueva, Codis no lineals en Magma: construcció de codis perfectes Universitat Autónoma de Barcelona, 2009. [22] M. Villanueva, F. Zeng and J. Pujol, Efficient representation of binary nonlinear codes: constructions and minimum distance computation, Des. Codes Crypt., 76 (2015), 3-21.  doi: 10.1007/s10623-014-0028-4. [23] J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575. [24] J. A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc., 136 (2008), 699-706.  doi: 10.1090/S0002-9939-07-09164-2.

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References:
 [1] T. Britz, MacWilliams identities and matroid polynomials, Electr. J. Combin., 9 (2002), R19, 16pp. [2] P. Delsarte, Bounds for unrestricted codes, by linear programming, Philips Res. Rep., 27 (1972), 272-289. [3] T. Etzion and A. Vardy, On perfect codes and tilings, problems and solutions, SIAM J. Discr. Math., 11 (1998), 205-223.  doi: 10.1137/S0895480196309171. [4] M. Greferath, An introduction to ring-linear coding theory, in Gröbner Bases, Coding and Cryptography (eds. M. Sala et al), Springer-Verlag, Berlin, 2009,219–238. [5] M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes, J. Alg. Appl., 3 (2004), 247-272.  doi: 10.1142/S0219498804000873. [6] M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams' equivalence theorem, J. Combin. Theory Ser. A, 92 (2000), 17-28.  doi: 10.1006/jcta.1999.3033. [7] A. R. Hammons, Jr., 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. Inf. Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154. [8] O. Heden, A full rank perfect code of length 31, Des. Codes Crypt., 38 (2006), 125-129.  doi: 10.1007/s10623-005-5665-1. [9] O. Heden, On perfect $p$-ary codes of length p+1, Des. Codes Crypt., 46 (2008), 45-56.  doi: 10.1007/s10623-007-9133-y. [10] O. Heden, Perfect codes from the dual point of view Ⅰ, Discr. Math., 308 (2008), 6141-6156.  doi: 10.1016/j.disc.2007.11.037. [11] M. Hessler, Perfect codes as isomorphic spaces, Discr. Math., 306 (2006), 1981-1987.  doi: 10.1016/j.disc.2006.03.039. [12] T. Honold, Characterization of finite Frobenius rings, Arch. Math., 76 (2001), 406-415.  doi: 10.1007/PL00000451. [13] T. Honold and A. A. Nechaev, Weighted modules and linear representations of codes, Probl. Inf. Transm., 35 (1999), 205-223. [14] T. Honold and I. Landjev, MacWilliams identities for linear codes over finite Frobenius rings, in Finite Fields and Applications (eds. D. Jungnickel et al), Springer-Verlag, Berlin, 2001, 276–292. [15] F. J. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell Sys. Tech. J., 42 (1963), 79-94.  doi: 10.1002/j.1538-7305.1963.tb04003.x. [16] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes North-Holland, Amsterdam, 1977. [17] A. A. Nechaev, Finite principal ideal rings, Mat. Sbornik, 20 (1973), 364-382. [18] A. A. Nechaev, Kerdock code in a cyclic form, Discr. Math. Appl., 1 (1991), 365-384.  doi: 10.1515/dma.1991.1.4.365. [19] R. Y. Sharp, Steps in Commutative Algebra 2nd edition, Cambridge Univ. Press, Cambridge, 2000. doi: 10.1017/CBO9780511626265. [20] A. Terras, Fourier Analysis on Finite Groups and Applications Cambridge Univ. Press, Cambridge, 1999. doi: 10.1017/CBO9780511626265. [21] M. Villanueva, Codis no lineals en Magma: construcció de codis perfectes Universitat Autónoma de Barcelona, 2009. [22] M. Villanueva, F. Zeng and J. Pujol, Efficient representation of binary nonlinear codes: constructions and minimum distance computation, Des. Codes Crypt., 76 (2015), 3-21.  doi: 10.1007/s10623-014-0028-4. [23] J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575. [24] J. A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc., 136 (2008), 699-706.  doi: 10.1090/S0002-9939-07-09164-2.
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