Abstract
In this work, we introduce the harmonic generalization of the m-tuple weight enumerators of codes over finite Frobenius rings. A harmonic version of the MacWilliams-type identity for m-tuple weight enumerators of codes over finite Frobenius ring is also given. Moreover, we define the demi-matroid analogue of well-known polynomials from matroid theory, namely Tutte polynomials and coboundary polynomials, and associate them with a harmonic function. We also prove the Greene-type identity relating these polynomials to the harmonic m-tuple weight enumerators of codes over finite Frobenius rings. As an application of this Greene-type identity, we provide a simple combinatorial proof of the MacWilliams-type identity for harmonic m-tuple weight enumerators over finite Frobenius rings. Finally, we provide the structure of the relative invariant spaces containing the harmonic m-tuple weight enumerators of self-dual codes over finite fields.
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References
Bachoc C.: On harmonic weight enumerators of binary codes. Des. Codes Cryptogr. 18(1–3), 11–28 (1999).
Bachoc C.: Harmonic weight enumerators of nonbinary codes and MacWilliams identities, Codes and Association Schemes (Piscataway, NJ, 1999), 1–23, DIMACS Series in Discrete Mathematics and Theoretical Computer 56. American Mathematical Society, Providence, RI, (2001).
Berlekamp E.R., MacWilliams F.J., Sloane N.J.A.: Gleason’s theorem on self-dual codes. IEEE Trans. Inform. Theory IT–18, 409–414 (1972).
Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997).
Britz T., Cameron P.J.: Codes. In: Ellis-Monaghan J.A., Moffatt I. (eds.) Handbook of the Tutte Polynomial and Related Topics, 1st edn, pp. 328–344. Chapman and Hall/CRC, Boca Raton (2022).
Britz T., Johnsen T., Mayhew D., Shiromoto K.: Wei-type duality theorems for matroids. Des. Codes Cryptogr. 62, 331–341 (2012).
Britz T., Shiromoto K.: A MacWilliams-type identity for matroids. Discret. Math. 308, 4551–4559 (2008).
Britz T., Shiromoto K., Westerbäck T.: Demi-matroids from codes over finite Frobenius rings. Des. Codes Cryptogr. 75, 97–107 (2015).
Chakraborty H.S., Miezaki T., Oura M.: Harmonic Tutte polynomials of matroids. Des. Codes Cryptogr. 91, 2223–2236 (2023).
Crapo H.H.: The Tutte polynomial. Aequ. Math. 3, 211–229 (1969).
Crapo H.H., Rota G.C.: On the Foundations of Combinatorial Theory: Combinatorial Geometries, Preliminary Edition. The M.I.T. Press, Cambridge (1970).
Conway J.H., Sloane N.J.A.: Sphere Packings Lattices and Groups, 3rd edn Springer, New York (1999).
Delsarte P.: Hahn polynomials, discrete harmonics, and \(t\)-designs. SIAM J. Appl. Math. 34, 157–166 (1978).
Gleason A.M.: Weight polynomials of self-dual codes and the MacWilliams identities. In: Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3, Gauthier-Villars, Paris, pp. 211–215 (1971).
Greene C.: Weight enumeration and the geometry of linear codes, Studia. Appl. Math. 55, 119–128 (1976).
Huffman W., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).
Karlin S., McGregor J.: The Hahn polynomials, formulas and an application. Scr. Math. 26, 33–46 (1961).
Lam T.Y.: Lectures on Modules and Rings. Springer, New York (1999).
MacWilliams F.J.: A theorem on the distribution of weights in a systematic code. Bell Syst. Tech. J. 42, 79–84 (1963).
MacWilliams F.J., Mallows C.L., Sloane N.J.A.: Generalizations of Gleason’s theorem on weight enumerators of self-dual codes. IEEE Trans. Inform. Theory IT–18, 794–805 (1972).
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes, North-Holland Editor (1977).
Miezaki T.: Miezaki’s, Tsuyoshi: website: http://www.f.waseda.jp/miezaki/data.html.
Molien T.: Über die Invarianten der linearen Substitutionsgruppen. Sitzungber. König. Preuss. Akad. Wiss. 52, 1152–1156 (1897).
Nebe G., Rains E.M., Sloane N.J.A.: Self-Dual Codes and Invariant Theory, Algorithms and Computation in Mathematics, vol. 17. Springer, Berlin (2006).
Shiromoto K.: A new MacWilliams-type identity for linear codes. Hokkaido Math. J. 25, 651–656 (1996).
Tanabe K.: A new proof of the Assmus-Mattson theorem for non-binary codes. Des. Codes Cryptogr. 22, 149–155 (2001).
Tutte W.T.: A contribution to the theory of chromatic polynomial. Can. J. Math. 6, 80–91 (1954).
Tutte W.T.: On dichromatic polynomials. J. Comb. Theory 2, 301–320 (1967).
Wei V.K.: Generalized Hamming weights for linear codes. IEEE Trans. Inf. Theory 37, 1412–1418 (1991).
Wolfram Research, Inc., Mathematica, Version 11.2, Champaign, IL (2017).
Acknowledgements
The authors would also like to thank the anonymous reviewers for their beneficial comments on an earlier version of the manuscript. The fourth named author is supported by JSPS KAKENHI (22K03277).
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Communicated by I. Landjev.
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Britz, T., Chakraborty, H.S., Ishikawa, R. et al. Harmonic Tutte polynomials of matroids II. Des. Codes Cryptogr. 92, 1279–1297 (2024). https://doi.org/10.1007/s10623-023-01343-0
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DOI: https://doi.org/10.1007/s10623-023-01343-0