May  2011, 5(2): 395-406. doi: 10.3934/amc.2011.5.395

On the weight distribution of codes over finite rings

1. 

School of Mathematical Sciences, University College Dublin, Springfield, MO 65801-2604, United States

Received  May 2010 Revised  November 2010 Published  May 2011

Let $R>S$ be finite Frobenius rings for which there exists a trace map $T:$ S$R \rightarrow$S$R$. Let $C$f,s$:=\{x \mapsto T(\alpha x + \beta f(x)) : \alpha, \beta \in R \}$. $C$f,s is an $S$-linear subring-subcode of a left linear code over $R$. We consider functions $f$ for which the homogeneous weight distribution of $C$f,s can be computed. In particular, we give constructions of codes over integer modular rings and commutative local Frobenius that have small spectra.
Citation: 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
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), 703-714. 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), 1-16. 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), 169-179. 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), 289-301. doi: 10.1007/s10623-006-9035-4.

[5]

E. Byrne and A. Sneyd, Constructions of two-weight codes over finite rings, in "Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2010),'' Budapest, July, 2010.

[6]

C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Crypt., 15 (1998), 125-156. 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), 2089-2013. 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), 22-28; translation in Problems Inform. Transmission, 33 (1997), 208-213.

[9]

P. Delsarte, Weights of linear codes and strongly regular normed spaces, Discrete Math., 3 (1972), 47-64. 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), 247-272. 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), 11-24. 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), 17-28. 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), 301-319. 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), 254-257. doi: 10.1109/TIT.1961.1057655.

[15]

T. Honold, Characterization of finite Frobenius rings, Arch. Math. (Basel), 76 (2001), 406-415.

[16]

T. Honold, Further results on homogeneous two-weight codes, in "Proceedings of Optimal Codes and Related Topics,'' Bulgaria, (2007).

[17]

T. Y. Lam, "Lectures on Modules and Rings,'' Springer-Verlag, 1999.

[18]

B. R. McDonald, Finite rings with identity, in "Pure and Applied Mathematics,'' Marcel Dekker, Inc., New York, (1974), 429.

[19]

R. Raghavendran, Finite associative rings, Compositio Math., 21 (1969), 195-229.

[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), 712-717. 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), 703-714. 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), 1-16. 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), 169-179. 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), 289-301. doi: 10.1007/s10623-006-9035-4.

[5]

E. Byrne and A. Sneyd, Constructions of two-weight codes over finite rings, in "Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2010),'' Budapest, July, 2010.

[6]

C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Crypt., 15 (1998), 125-156. 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), 2089-2013. 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), 22-28; translation in Problems Inform. Transmission, 33 (1997), 208-213.

[9]

P. Delsarte, Weights of linear codes and strongly regular normed spaces, Discrete Math., 3 (1972), 47-64. 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), 247-272. 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), 11-24. 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), 17-28. 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), 301-319. 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), 254-257. doi: 10.1109/TIT.1961.1057655.

[15]

T. Honold, Characterization of finite Frobenius rings, Arch. Math. (Basel), 76 (2001), 406-415.

[16]

T. Honold, Further results on homogeneous two-weight codes, in "Proceedings of Optimal Codes and Related Topics,'' Bulgaria, (2007).

[17]

T. Y. Lam, "Lectures on Modules and Rings,'' Springer-Verlag, 1999.

[18]

B. R. McDonald, Finite rings with identity, in "Pure and Applied Mathematics,'' Marcel Dekker, Inc., New York, (1974), 429.

[19]

R. Raghavendran, Finite associative rings, Compositio Math., 21 (1969), 195-229.

[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), 712-717. doi: 10.1109/TIT.2005.862125.

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