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On the weight distribution of codes over finite rings

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  • 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.
    Mathematics Subject Classification: Primary: 11T71; Secondary: 14G50.


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  • [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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


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


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


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


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


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


    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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