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Codes from the incidence matrices and line graphs of Hamming graphs $H^k(n,2)$ for $k \geq 2$
On the weight distribution of codes over finite rings
1. | School of Mathematical Sciences, University College Dublin, Springfield, MO 65801-2604, United States |
References:
[1] |
C. Bracken, E. Byrne, N. Markin and G. McGuire, New families of almost perfect nonlinear trinomials and multinomials,, Finite Fields Appl., 14 (2008), 703.
doi: 10.1016/j.ffa.2007.11.002. |
[2] |
E. Byrne, M. Greferath and T. Honold, Ring geometries, two-weight codes and strongly regular graphs,, Des. Codes Crypt., 48 (2008), 1.
doi: 10.1007/s10623-007-9136-8. |
[3] |
E. Byrne, M. Greferath, A. Kohnert and V. Skachek, New bounds for codes over finite Frobenius rings,, Des. Codes Crypt., 57 (2010), 169.
doi: 10.1007/s10623-009-9359-y. |
[4] |
E. Byrne, M. Greferath and M. E. O'Sullivan, The linear programming bound for codes over finite Frobenius rings,, Des. Codes Crypt., 42 (2007), 289.
doi: 10.1007/s10623-006-9035-4. |
[5] |
E. Byrne and A. Sneyd, Constructions of two-weight codes over finite rings,, in, (2010). Google Scholar |
[6] |
C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems,, Des. Codes Crypt., 15 (1998), 125.
doi: 10.1023/A:1008344232130. |
[7] |
C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes,, IEEE Trans. Inform. Theory, 51 (2005), 2089.
doi: 10.1109/TIT.2005.847722. |
[8] |
I. Constantinescu and W. Heise, A metric for codes over residue class rings of integers (in Russian),, Problemy Peredachi Informatsii, 33 (1997), 22.
|
[9] |
P. Delsarte, Weights of linear codes and strongly regular normed spaces,, Discrete Math., 3 (1972), 47.
doi: 10.1016/0012-365X(72)90024-6. |
[10] |
M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes,, J. Algebra Appl., 3 (2004), 247.
doi: 10.1142/S0219498804000873. |
[11] |
M. Greferath and M. E. O'Sullivan, On bounds for codes over Frobenius rings under homogeneous weights,, Discrete Math., 289 (2004), 11.
doi: 10.1016/j.disc.2004.10.002. |
[12] |
M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams equivalence theorem,, J. Combin. Theory A, 92 (2000), 17.
doi: 10.1006/jcta.1999.3033. |
[13] |
A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbbZ_4$-linearity of Kerdock, Preparata, Goethals and related codes,, IEEE Trans. Inform. Theory, 40 (1994), 301.
doi: 10.1109/18.312154. |
[14] |
R. C. Heimiller, Phase shift pulse codes with good periodic correlation properties,, IRE Trans. Inform. Theory, IT-7 (1961), 254.
doi: 10.1109/TIT.1961.1057655. |
[15] |
T. Honold, Characterization of finite Frobenius rings,, Arch. Math. (Basel), 76 (2001), 406.
|
[16] |
T. Honold, Further results on homogeneous two-weight codes,, in, (2007). Google Scholar |
[17] |
T. Y. Lam, "Lectures on Modules and Rings,'', Springer-Verlag, (1999).
|
[18] |
B. R. McDonald, Finite rings with identity,, in, (1974).
|
[19] |
R. Raghavendran, Finite associative rings,, Compositio Math., 21 (1969), 195.
|
[20] |
J. Yuan, C. Carlet and C. Ding, The weight distribution of a class of linear codes from perfect nonlinear functions,, IEEE Trans. Inform. Theory, 52 (2006), 712.
doi: 10.1109/TIT.2005.862125. |
show all references
References:
[1] |
C. Bracken, E. Byrne, N. Markin and G. McGuire, New families of almost perfect nonlinear trinomials and multinomials,, Finite Fields Appl., 14 (2008), 703.
doi: 10.1016/j.ffa.2007.11.002. |
[2] |
E. Byrne, M. Greferath and T. Honold, Ring geometries, two-weight codes and strongly regular graphs,, Des. Codes Crypt., 48 (2008), 1.
doi: 10.1007/s10623-007-9136-8. |
[3] |
E. Byrne, M. Greferath, A. Kohnert and V. Skachek, New bounds for codes over finite Frobenius rings,, Des. Codes Crypt., 57 (2010), 169.
doi: 10.1007/s10623-009-9359-y. |
[4] |
E. Byrne, M. Greferath and M. E. O'Sullivan, The linear programming bound for codes over finite Frobenius rings,, Des. Codes Crypt., 42 (2007), 289.
doi: 10.1007/s10623-006-9035-4. |
[5] |
E. Byrne and A. Sneyd, Constructions of two-weight codes over finite rings,, in, (2010). Google Scholar |
[6] |
C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems,, Des. Codes Crypt., 15 (1998), 125.
doi: 10.1023/A:1008344232130. |
[7] |
C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes,, IEEE Trans. Inform. Theory, 51 (2005), 2089.
doi: 10.1109/TIT.2005.847722. |
[8] |
I. Constantinescu and W. Heise, A metric for codes over residue class rings of integers (in Russian),, Problemy Peredachi Informatsii, 33 (1997), 22.
|
[9] |
P. Delsarte, Weights of linear codes and strongly regular normed spaces,, Discrete Math., 3 (1972), 47.
doi: 10.1016/0012-365X(72)90024-6. |
[10] |
M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes,, J. Algebra Appl., 3 (2004), 247.
doi: 10.1142/S0219498804000873. |
[11] |
M. Greferath and M. E. O'Sullivan, On bounds for codes over Frobenius rings under homogeneous weights,, Discrete Math., 289 (2004), 11.
doi: 10.1016/j.disc.2004.10.002. |
[12] |
M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams equivalence theorem,, J. Combin. Theory A, 92 (2000), 17.
doi: 10.1006/jcta.1999.3033. |
[13] |
A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbbZ_4$-linearity of Kerdock, Preparata, Goethals and related codes,, IEEE Trans. Inform. Theory, 40 (1994), 301.
doi: 10.1109/18.312154. |
[14] |
R. C. Heimiller, Phase shift pulse codes with good periodic correlation properties,, IRE Trans. Inform. Theory, IT-7 (1961), 254.
doi: 10.1109/TIT.1961.1057655. |
[15] |
T. Honold, Characterization of finite Frobenius rings,, Arch. Math. (Basel), 76 (2001), 406.
|
[16] |
T. Honold, Further results on homogeneous two-weight codes,, in, (2007). Google Scholar |
[17] |
T. Y. Lam, "Lectures on Modules and Rings,'', Springer-Verlag, (1999).
|
[18] |
B. R. McDonald, Finite rings with identity,, in, (1974).
|
[19] |
R. Raghavendran, Finite associative rings,, Compositio Math., 21 (1969), 195.
|
[20] |
J. Yuan, C. Carlet and C. Ding, The weight distribution of a class of linear codes from perfect nonlinear functions,, IEEE Trans. Inform. Theory, 52 (2006), 712.
doi: 10.1109/TIT.2005.862125. |
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