Abstract
Relative one-weight linear codes were introduced by Liu and Chen over finite fields. These codes can be defined just as simply for egalitarian and homogeneous weights over Frobenius bimodule alphabets. A key lemma helps describe the structure of relative one-weight codes, and certain known types of two-weight linear codes can then be constructed easily. The key lemma also provides another approach to the MacWilliams extension theorem.
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References
Assmus Jr, E.F., Mattson Jr, H.F.: Error-correcting codes: an axiomatic approach. Inform. Control 6, 315–330 (1963).
Bogart K., Goldberg D., Gordon J.: An elementary proof of the MacWilliams theorem on equivalence of codes. Inform. Control 37(1), 19–22 (1978).
Calderbank A.R., Kantor W.M.: The geometry of two-weight codes. Bull. London Math. Soc. 18, 97–122 (1986).
Camion P.: Codes and association schemes: basic properties of association schemes relevant to coding. In: Handbook of Coding Theory, vols. I, II, pp. 1441–1566. North-Holland, Amsterdam (1998).
Chen W., Kløve T.: The weight hierarchies of \(q\)-ary codes of dimension \(4\). IEEE Trans. Inform. Theory 42 (6, part 2), 2265–2272 (1996).
Constantinescu I., Heise W.: On the concept of code-isomorphy. J. Geom. 57(1–2), 63–69 (1996).
Greferath M.: Orthogonality matrices for modules over finite Frobenius rings and MacWilliams’ equivalence theorem. Finite Fields Appl. 8(3), 323–331 (2002).
Greferath M., Nechaev A., Wisbauer R.: Finite quasi-Frobenius modules and linear codes. J. Algebra Appl. 3(3), 247–272 (2004).
Greferath M., Schmidt S.E.: Finite-ring combinatorics and MacWilliams’s equivalence theorem. J. Combin. Theory Ser. A 92(1), 17–28 (2000).
Heise W., Honold T.: Homogeneous and egalitarian weights on finite rings. In: Proceedings of the Seventh International Workshop on Algebraic and Combinatorial Coding Theory (ACCT-2000), pp. 183–188. Bansko, Bulgaria (2000).
Hirano Y.: On admissible rings. Indag. Math. (N.S.) 8(1), 55–59 (1997).
Honold T.: Characterization of finite Frobenius rings. Arch. Math. (Basel) 76(6), 406–415 (2001).
Honold T.: Further results on homogeneous two-weight codes. Proceedings of Optimal Codes and Related Topics. Bulgaria (2007).
Honold T., Landjev I.: Linear codes over finite chain rings. Electron. J. Combin. 7, Research Paper 11, 22 (2000). http://www.combinatorics.org/Volume_7/Abstracts/v7i1r11.html.
Honold T., Nechaev A.A.: Weighted modules and representations of codes. Problems Inform. Transm. 35(3), 205–223 (1999).
Lam T.Y.: Lectures on modules and rings, Graduate Texts in Math., vol. 189. Springer, New York (1999).
Liu Z., Chen W.: Notes on the value function. Des. Codes Cryptogr. 54(1), 11–19 (2010).
Liu Z., Chen W., Sun Z., Zeng X.: Further results on support weights of certain subcodes. Des. Codes Cryptogr. 61(2), 119–129 (2011).
MacWilliams F.J.: Error-correcting codes for multiple-level transmission. Bell System Tech. J. 40, 281–308 (1961).
Peterson W.W.: Error-Correcting Codes. The MIT Press, Cambridge (1961).
Stanley R.P.: Enumerative combinatorics, The Wadsworth& Brooks/Cole Mathematics Series, vol. I. Wadsworth& Brooks/Cole Advanced Books& Software, Monterey, CA (1986).
Terras A.: Fourier analysis on finite groups and applications, London Mathematical Society Student Texts, vol. 43. Cambridge University Press, Cambridge (1999).
Tsfasman M.A., Vlăduţ S.G.: Algebraic-geometric codes. In: Mathematics and its Applications (Soviet Series), vol. 58. Kluwer Academic Publishers Group, Dordrecht (1991).
Wood J.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121(3), 555–575 (1999).
Wood J.A.: The structure of linear codes of constant weight. Trans. Am. Math. Soc. 354(3), 1007–1026 (2002).
Wood J.A.: Foundations of linear codes defined over finite modules: the extension theorem and the MacWilliams identities. In: Codes Over Rings (Ankara, 2008), Ser. Coding Theory Cryptol., vol. 6, pp. 124–190. World Sci. Publ., Hackensack (2009).
Wood J.A.: Applications of finite Frobenius rings to the foundations of algebraic coding theory. In: Iyama, O. (ed.) Proceedings of the 44th Symposium on Rings and Representation Theory (Okayama University, Japan, September 25–27, 2011), pp. 223–245. Nagoya (2012).
Acknowledgments
I thank the Department of Mathematics, Huazhong Normal University, Wuhan, China, and especially Professors Yun Fan and Hongwei Liu, for their hospitality during the summer of 2011, when much of the research for this paper was conducted. I thank the referees for their helpful comments, especially those related to the history of homogeneous weights, and for saving me from some embarassing typos. I also thank my wife Elizabeth S. Moore for her continuing support and encouragement. This work was partially supported by a sabbatical leave from Western Michigan University.
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Communicated by R. Hill.
In memory of Professor F. E. P. Hirzebruch, 17 October 1927–27 May 2012.
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Wood, J.A. Relative one-weight linear codes. Des. Codes Cryptogr. 72, 331–344 (2014). https://doi.org/10.1007/s10623-012-9769-0
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DOI: https://doi.org/10.1007/s10623-012-9769-0