-
Previous Article
Complete characterization of the first descent point distribution for the k-error linear complexity of 2n-periodic binary sequences
- AMC Home
- This Issue
- Next Article
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 |
The concept of parity check matrices of linear binary codes has been extended by Heden [
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. |
show all references
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. |
[1] |
Emily McMillon, Allison Beemer, Christine A. Kelley. Extremal absorbing sets in low-density parity-check codes. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021003 |
[2] |
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 |
[3] |
Ferruh Özbudak, Patrick Solé. Gilbert-Varshamov type bounds for linear codes over finite chain rings. Advances in Mathematics of Communications, 2007, 1 (1) : 99-109. doi: 10.3934/amc.2007.1.99 |
[4] |
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 |
[5] |
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 |
[6] |
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 |
[7] |
Nuh Aydin, Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Esengül Saltürk. Skew constacyclic codes over the local Frobenius non-chain rings of order 16. Advances in Mathematics of Communications, 2020, 14 (1) : 53-67. doi: 10.3934/amc.2020005 |
[8] |
Gianira N. Alfarano, Anina Gruica, Julia Lieb, Joachim Rosenthal. Convolutional codes over finite chain rings, MDP codes and their characterization. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022028 |
[9] |
Igor E. Shparlinski. On some dynamical systems in finite fields and residue rings. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 901-917. doi: 10.3934/dcds.2007.17.901 |
[10] |
Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure and Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565 |
[11] |
Heide Gluesing-Luerssen. Partitions of Frobenius rings induced by the homogeneous weight. Advances in Mathematics of Communications, 2014, 8 (2) : 191-207. doi: 10.3934/amc.2014.8.191 |
[12] |
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 |
[13] |
Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schrödinger operators. Inverse Problems and Imaging, 2014, 8 (2) : 475-489. doi: 10.3934/ipi.2014.8.475 |
[14] |
Amina Amassad, Mircea Sofonea. Analysis of some nonlinear evolution systems arising in rate-type viscoplasticity. Conference Publications, 1998, 1998 (Special) : 58-71. doi: 10.3934/proc.1998.1998.58 |
[15] |
Anderson Silva, C. Polcino Milies. Cyclic codes of length $ 2p^n $ over finite chain rings. Advances in Mathematics of Communications, 2020, 14 (2) : 233-245. doi: 10.3934/amc.2020017 |
[16] |
Zilong Wang, Guang Gong. Correlation of binary sequence families derived from the multiplicative characters of finite fields. Advances in Mathematics of Communications, 2013, 7 (4) : 475-484. doi: 10.3934/amc.2013.7.475 |
[17] |
Genady Ya. Grabarnik, Misha Guysinsky. Livšic theorem for banach rings. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4379-4390. doi: 10.3934/dcds.2017187 |
[18] |
Michael Ruzhansky, Jens Wirth. Dispersive type estimates for fourier integrals and applications to hyperbolic systems. Conference Publications, 2011, 2011 (Special) : 1263-1270. doi: 10.3934/proc.2011.2011.1263 |
[19] |
Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035 |
[20] |
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 |
2020 Impact Factor: 0.935
Tools
Metrics
Other articles
by authors
[Back to Top]