On the weight distribution of codes over finite rings

Let R > S be finite Frobenius rings for which there exists a trace map T from R onto S as left S modules. Let C:= {x -> T(ax + bf(x)) : a,b in R}. Then C 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 can be computed. In particular, we give constructions of codes over integer modular rings and commutative local Frobenius that have small spectra.

[1]  Marcus Greferath,et al.  On bounds for codes over Frobenius rings under homogeneous weights , 2004, Discret. Math..

[2]  Eimear Byrne,et al.  New bounds for codes over finite Frobenius rings , 2010, Des. Codes Cryptogr..

[3]  Tsit Yuen Lam,et al.  Lectures on modules and rings , 1998 .

[4]  Claude Carlet,et al.  Codes, Bent Functions and Permutations Suitable For DES-like Cryptosystems , 1998, Des. Codes Cryptogr..

[5]  T. Honold,et al.  Characterization of finite Frobenius rings , 2001 .

[6]  Marcus Greferath,et al.  Finite-Ring Combinatorics and MacWilliams' Equivalence Theorem , 2000, J. Comb. Theory A.

[7]  D. G. Glynn Ring of geometries I , 1987 .

[8]  R. Raghavendran,et al.  Finite associative rings , 1969 .

[9]  R. C. Heimiller,et al.  Phase shift pulse codes with good periodic correlation properties , 1961, IRE Trans. Inf. Theory.

[10]  Eimear Byrne,et al.  New families of quadratic almost perfect nonlinear trinomials and multinomials , 2008, Finite Fields Their Appl..

[11]  N. J. A. Sloane,et al.  The Z4-linearity of Kerdock, Preparata, Goethals, and related codes , 1994, IEEE Trans. Inf. Theory.

[12]  B. R. McDonald Finite Rings With Identity , 1974 .

[13]  Eimear Byrne,et al.  Constructions of Two-Weight Codes Over Finite Rings , 2010 .

[14]  Cunsheng Ding,et al.  The weight distribution of a class of linear codes from perfect nonlinear functions , 2006, IEEE Transactions on Information Theory.

[15]  A. A. Nechaev,et al.  FINITE QUASI-FROBENIUS MODULES AND LINEAR CODES , 2004 .

[16]  Philippe Delsarte,et al.  Weights of linear codes and strongly regular normed spaces , 1972, Discret. Math..

[17]  Eimear Byrne,et al.  The linear programming bound for codes over finite Frobenius rings , 2007, Des. Codes Cryptogr..

[18]  Cunsheng Ding,et al.  Linear codes from perfect nonlinear mappings and their secret sharing schemes , 2005, IEEE Transactions on Information Theory.

[19]  Eimear Byrne,et al.  Ring geometries, two-weight codes, and strongly regular graphs , 2007, Des. Codes Cryptogr..