\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

($\sigma,\delta$)-codes

Abstract Related Papers Cited by
  • In this paper we introduce the notion of cyclic ($f(t),\sigma,\delta$)-codes for $f(t)\in A[t;\sigma,\delta]$. These codes generalize the $\theta$-codes as introduced by D. Boucher, F. Ulmer, W. Geiselmann [2]. We construct generic and control matrices for these codes. As a particular case the ($\sigma,\delta$)-$W$-code associated to a Wedderburn polynomial are defined and we show that their control matrices are given by generalized Vandermonde matrices. All the Wedderburn polynomials of $\mathbb F_q[t;\theta]$ are described and their control matrices are presented. A key role will be played by the pseudo-linear transformations.
    Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 16S36.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    S. A. Amitsur, Derivations in simple rings, Proc. London Math. Soc., 3 (1957), 87-112.

    [2]

    D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engin. Commun. Comp., 18 (2007), 379-389.doi: 10.1007/s00200-007-0043-z.

    [3]

    D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.doi: 10.3934/amc.2008.2.273.

    [4]

    D. Boucher and F. UlmerLinear codes using skew polynomials with automorphisms and derivations, Des. Codes Cryptogr., to appear. doi: 10.1007/s10623-012-9704-4.

    [5]

    J. Delenclos and A. Leroy, Noncommutative symmetric functions and W-polynomials, J. Algebra Appl., 6 (2007), 815-837.doi: 10.1142/S021949880700251X.

    [6]

    M. Giesbrecht, Factoring in skew polynomial rings over finite fields, J. Symb. Comp., 26 (1998), 463-468.doi: 10.1006/jsco.1998.0224.

    [7]

    N. Jacobson, On pseudo linear transformations, Ann. Math., 38 (1937), 484-507.doi: 10.2307/1968565.

    [8]

    S. K. Jain and S. R. Nagpaul, Topics in Applied Abstract Algebra, AMS, 2005.

    [9]

    T. Y. Lam and A. Leroy, Wedderburn polynomials over division rings, I, J. Pure Appl. Algebra, 186 (2004), 43-76.doi: 10.1016/S0022-4049(03)00125-7.

    [10]

    T. Y. Lam, A. Leroy and A. Ozturk, Wedderburn polynomial over division rings, II, Contemp. Math., 456 (2008), 73-98.doi: 10.1090/conm/456/08885.

    [11]

    A. Leroy, Pseudo-linear transformation and evaluation in Ore extension, Bull. Belg. Math. Soc., 2 (1995), 321-345.

    [12]

    A. Leroy, Noncommutative polynomial maps, J. Algebra Appl., 11 (2012).doi: 10.1142/S0219498812500764.

    [13]

    S. R. López-Permouth and S. Szabo, Convolutional codes with additional algebraic structures, J. Pure Appl. Algebra, (2012).doi: 10.1016/j.jpaa.2012.09.017.

    [14]

    F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North-Holland, Amsterdam, 1978.

    [15]

    P. Solé, Codes over rings, in Proceeding of the CIMPA Summer School, Ankara, Turkey, 2008.doi: 10.1109/TIT.2013.2277721.

    [16]

    P. Solé and O. Yemen, Binary quasi-cyclic codes of index 2 and skew polynomial rings, Finite Fields Appl., 18 (2012), 685-699.doi: 10.1016/j.ffa.2012.02.002.

    [17]

    J. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc., 136 (2008), 699-706.doi: 10.1090/S0002-9939-07-09164-2.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(68) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return