Abstract
The extension problem underlies notions of code equivalence. Two approaches to the extension problem are described. One is a matrix approach that reduces the general problem for weights to one for symmetrized weight compositions. The other is a monoid algebra approach that reframes the extension problem in terms of modules over the monoid algebra determined by the multiplicative monoid of a finite ring.
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Barra, A.: Equivalence Theorems and the Local-Global Property. ProQuest LLC, Ann Arbor, MI (2012). http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:3584023. Thesis (Ph.D.)–University of Kentucky
Barra, A., Gluesing-Luerssen, H.: MacWilliams extension theorems and the local-global property for codes over Frobenius rings. J. Pure Appl. Algebra 219(4), 703–728 (2015). https://doi.org/10.1016/j.jpaa.2014.04.026
Clark, A.E., Drake, D.A.: Finite chain rings. Abh. Math. Sem. Univ. Hamb. 39, 147–153 (1973)
Dyshko, S.: The extension theorem for Lee and Euclidean weight codes over integer residue rings. Des. Codes Cryptogr. 87(6), 1253–1269 (2019). https://doi.org/10.1007/s10623-018-0521-2
Dyshko, S., Langevin, P., Wood, J.A.: Deux analogues au déterminant de Maillet. C. R. Math. Acad. Sci. Paris 354(7), 649–652 (2016)
ElGarem, N., Megahed, N., Wood, J.A.: The extension theorem with respect to symmetrized weight compositions. In: Pinto, R., Malonek, P.R., Vettori, P. (eds.) Coding Theory and Applications. CIM Series in Mathematical Sciences, vol. 3, pp. 177–183. Springer, Berlin (2015)
Gnilke, O.W., Greferath, M., Honold, T., Wood, J.A., Zumbrägel, J.: The extension theorem for bi-invariant weights over Frobenius rings and Frobenius bimodules. In: Leroy, A., Lomp, C., López-Permouth, S., Oggier, F. (eds.) Rings, Modules and Codes, Contemporary Mathematics, vol. 727, pp. 117–129. Amer. Math. Soc., Providence, RI (2019)
Greferath, M., Honold, T.: Monomial extensions of isometries of linear codes II: Invariant weight functions on \(Z_m\). In: Proceedings of the Tenth International Workshop in Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 106–111. Zvenigorod, Russia (2006)
Greferath, M., Honold, T., Mc Fadden, C., Wood, J.A., Zumbrägel, J.: MacWilliams’ extension theorem for bi-invariant weights over finite principal ideal rings. J. Combin. Theory Ser. A 125, 177–193 (2014). https://doi.org/10.1016/j.jcta.2014.03.005
Greferath, M., Mc Fadden, C., Zumbrägel, J.: Characteristics of invariant weights related to code equivalence over rings. Des. Codes Cryptogr. 66(1–3), 145–156 (2013). https://doi.org/10.1007/s10623-012-9671-9
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)
Honold, T.: Characterization of finite Frobenius rings. Arch. Math. (Basel) 76(6), 406–415 (2001)
Honold, T., Nechaev, A.A.: Weighted modules and representations of codes. Problems Inform. Transmiss. 35(3), 205–223 (1999)
Langevin, P., Wood, J.A.: The extension problem for Lee and Euclidean weights. J. Algebra Comb. Discrete Struct. Appl. 4(2), 207–217 (2017). https://doi.org/10.13069/jacodesmath.284970
Langevin, P., Wood, J.A.: The extension theorem for the Lee and Euclidean weights over \(\mathbb{{Z}}/p^k\mathbb{{Z}}\). J. Pure Appl. Algebra 223, 922–930 (2019)
MacWilliams, F.J.: Error-correcting codes for multiple-level transmission. Bell Syst. Tech. J. 40, 281–308 (1961)
MacWilliams, F.J.: Combinatorial problems of elementary abelian groups. Ph.D. thesis, Radcliffe College, Cambridge, MA (1962)
Wood, J.A.: Extension theorems for linear codes over finite rings. In: Mora, T., Mattson, H. (eds.) Applied Algebra, Algebraic Algorithms and Error-correcting Codes (Toulouse, 1997). Lecture Notes in Comput. Sci., vol. 1255, pp. 329–340. Springer, Berlin (1997)
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.: Weight functions and the extension theorem for linear codes over finite rings. In: Mullin, R.C., Mullen, G.L. (eds.) Finite Fields: Theory, Applications, and Algorithms (Waterloo, ON, 1997). Contemp. Math., vol. 225, pp. 231–243. Amer. Math. Soc., Providence, RI (1999)
Wood, J.A.: Factoring the semigroup determinant of a finite chain ring. In: Buchmann, J., Holdt, T.H., Stichtenoth, H., Tapia-Recillas, H. (eds.) Coding Theory, Cryptography and Related Areas, pp. 249–259. Springer, Berlin (2000)
Wood, J.A.: Foundations of linear codes defined over finite modules: the extension theorem and the MacWilliams identities. In: Solé, P. (ed.) Codes over rings (Ankara, 2008). Ser. Coding Theory Cryptol., vol. 6, pp. 124–190. World Sci. Publ, Hackensack, NJ (2009)
Wood, J.A.: Applications of finite Frobenius rings to the foundations of algebraic coding theory. In: Proceedings of the 44th Symposium on Ring Theory and Representation Theory, pp. 223–245. Symp. Ring Theory Represent. Theory Organ. Comm., Nagoya (2012)
Acknowledgements
I thank Noha Abdelghany for a number of conversations that helped me clarify various ideas, such as Theorem 4.5. I thank the anonymous referees for catching several typos and suggesting some clarification in the proof of Proposition 5.4. I thank my wife Elizabeth S. Moore for her steadfast encouragement and support, especially while I was trying to prove Theorem 4.3. This paper is dedicated to the memory of our mothers.
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In memoriam: Florence E. Wood, 1922–2019. Mary Elizabeth Keyte Moore, 1932–2019.
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Wood, J.A. Two approaches to the extension problem for arbitrary weights over finite module alphabets. AAECC 32, 427–455 (2021). https://doi.org/10.1007/s00200-020-00465-5
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DOI: https://doi.org/10.1007/s00200-020-00465-5