Advanced Search
Article Contents
Article Contents

On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes

Abstract Related Papers Cited by
  • In [7], self-orthogonal additive codes over $\mathbb{F}_4$ under the trace inner product were connected to binary quantum codes; a similar connection was given in the nonbinary case in [33]. In this paper we consider a natural generalization of additive codes called $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. We examine a number of classical results from the theory of $\mathbb{F}_q$-linear codes, and see how they must be modified to give analogous results for $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Included in the topics examined are the MacWilliams Identities, the Gleason polynomials, the Gleason-Pierce Theorem, Mass Formulas, the Balance Principle, the Singleton Bound, and MDS codes. We also classify certain of these codes for small lengths using the theory developed.
    Mathematics Subject Classification: Primary: 94B60, 94B05; Secondary: 94B27.


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

    I. M. Araújo, et al.GAP Reference Manual, The GAP Group, http://www.gap-system.org


    E. F. Assmus, Jr., H. F. Mattson, Jr. and R. J. Turyn, Research to develop the algebraic theory of codes, Report AFCRL-67-0365, Air Force Cambridge Res. Labs., Bedford, MA, 1967.


    C. Bachoc and P. Gaborit, On extremal additive $GF(4)$-codes of lengths $10$ to $18$, J. Théorie Nombres Bordeaux, 12 (2000), 225-272.doi: 10.5802/jtnb.278.


    J. Bierbrauer, Cyclic additive and quantum stabilizer codes, in "Arithmetic of Finite Fields: First International Workshop'' (eds. C. Carlet and B. Sunar), Madrid, (2007), 276-283.doi: 10.1007/978-3-540-73074-3_21.


    J. Bierbrauer and Y. Edel, Quantum twisted codes, J. Combin. Des., 8 (2000), 174-188.doi: 10.1002/(SICI)1520-6610(2000)8:3<174::AID-JCD3>3.0.CO;2-T.


    G. Birkhoff and S. MacLane, "A Survey of Modern Algebra,'' $4^{th}$ edition, MacMillan Publishing, New York, 1977.


    A. R. Calderbank, E. M. Rains, P. M. Shor and N. J. A. Sloane, Quantum error correction via codes over $GF(4)$, IEEE Trans. Inform. Theory, IT-44 (1998), 1369-1387.doi: 10.1109/18.681315.


    L. E. Danielsen, Graph-based classification of self-dual additive codes over finite fields, Adv. Math. Commun., 3 (2009), 329-348.doi: 10.3934/amc.2009.3.329.


    L. E. Danielsen, On the classification of Hermitian self-dual additive codes over $GF(9)$, IEEE Trans. Inform. Theory, IT-58 (2012), 5500-5511.doi: 10.1109/TIT.2012.2196255.


    L. E. Danielsen and M. G. Parker, On the classification of all self-dual additive codes over $GF(4)$ of length up to $12$, J. Comb. Theory Ser. A, 113 (2006), 1351-1367.doi: 10.1016/j.jcta.2005.12.004.


    B. K. Dey and B. S. Rajan, $\mathbb F_q$-linear cyclic codes over $\mathbb F_{q^m}$: DFT approach, Des. Codes Cryptogr., 34 (2005), 89-116.doi: 10.1007/s10623-003-4196-x.


    L. Dornhoff, "Group Representation Theory (Part A),'' Marcel Dekker, New York, 1971.


    C. Drees, M. Epkenhans and M. Krüskemper, On the computation of the trace form of some Galois extensions, J. Algebra, 192 (1997), 209-234.doi: 10.1006/jabr.1996.6939.


    J. E. Fields, P. Gaborit, W. C. Huffman and V. Pless, On the classification of extremal even formally self-dual codes, Des. Codes Cryptogr., 18 (1999), 125-148.doi: 10.1023/A:1008389220478.


    J. E. Fields, P. Gaborit, W. C. Huffman and V. Pless, On the classification of extremal even formally self-dual codes of lengths $20$ and $22$, Discrete Appl. Math., 111 (2001), 75-86.doi: 10.1016/S0166-218X(00)00345-0.


    P. Gaborit, W. C. Huffman, J.-L. Kim and V. Pless, On additive $GF(4)$ codes, in "Codes and Association Schemes: DIMACS Workshop'' (eds. A. Barg and S. Litsyn), Amer. Math. Soc., Providence, (2001), 135-149.


    A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, Actes Congrés Internl. de Mathématique, Gauthier-Villars, Paris, (1971), 211-215.


    G. Höhn, Self-dual codes over the Kleinian four group, Math. Ann., 327 (2003), 227-255.doi: 10.1007/s00208-003-0440-y.


    S. K. Houghten, C. W. H. Lam and L. H. Thiel, Construction of $(48,24,12)$ doubly-even self-dual codes, Congr. Numer., 103 (1994), 41-53.


    S. K. Houghten, C. W. H. Lam, L. H. Thiel and J. A. Parker, The extended quadratic residue code is the only $(48,24,12)$ self-dual doubly-even code, IEEE Trans. Inform. Theory, IT-49 (2003), 53-59.doi: 10.1109/TIT.2002.806146.


    W. C. Huffman, Additive cyclic codes over $\mathbb F_4$, Adv. Math. Commun., 1 (2007), 429-461.doi: 10.3934/amc.2007.1.427.


    W. C. Huffman, Additive cyclic codes over $\mathbb F_4$ of even length, Adv. Math. Commun., 2 (2008), 309-343.doi: 10.3934/amc.2008.2.309.


    W. C. Huffman, Cyclic $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes, Int. J. Inf. Coding Theory, 1 (2010), 249-284.doi: 10.1504/IJICOT.2010.032543.


    W. C. Huffman, Self-dual $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes with an automorphism of prime order, Adv. Math. Commun., 7 (2013), 57-90.doi: 10.3934/amc.2013.7.57.


    W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003.


    J.-L. Kim and X. Liu, A generalized Gleason-Pierce-Ward theorem, Des. Codes Cryptogr., 52 (2009), 363-380.doi: 10.1007/s10623-009-9286-y.


    J.-L. Kim and J. Walker, Nonbinary quantum error-correcting codes from algebraic curves, Discrete Math., 308 (2008), 3115-3124.doi: 10.1016/j.disc.2007.08.038.


    H. Koch, Unimodular lattices and self-dual codes, in "Proc. Intern. Congress Math.,'' Amer. Math. Soc., Providence, (1987), 457-465.


    T. Y. Lam, "The Algebraic Theory of Quadratic Forms,'' Reading MA: WA Benjamin, 1973.


    F. J. MacWilliams, A theorem on the distribution of weights in a systematic code, Bell System Tech. J., 42 (1963), 79-94.


    F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' Elsevier, New York, 1977.


    V. Pless, Power moment identities on weight distributions in error correcting codes, Inform. Control, 6 (1963), 147-152.doi: 10.1016/S0019-9958(63)90189-X.


    E. M. Rains, Nonbinary quantum codes, IEEE Trans. Inform. Theory, IT-45 (1999), 1827-1832.doi: 10.1109/18.782103.


    E. M. Rains and N. J. A. Sloane, Self-dual codes, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 177-294.


    J. J. Rotman, "An Introduction to the Theory of Groups,'' $4^{th}$ edition, Springer-Verlag, New York, 1995.doi: 10.1007/978-1-4612-4176-8.


    N. J. A. Sloane, Relations between combinatorics and other parts of mathematics, Proc. Sympos. Pure Math., 34 (1979), 273-308.


    D. E. Taylor, "The Geometry of the Classical Groups,'' Heldermann Verlag, Berlin, 1992.


    H. N. Ward, Divisible codes, Arch. Math. (Basel), 36 (1981), 485-494.doi: 10.1007/BF01223730.


    H. N. Ward, Quadratic residue codes and divisibility, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 827-870.

  • 加载中

Article Metrics

HTML views() PDF downloads(148) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint