Construction of self-dual binary \([2^{2k},2^{2k-1},2^k]\)-codes

Carolin Hannusch, Piroska Lakatos


The binary Reed-Muller code \({\rm RM}(m-k,m)\) corresponds to the \(k\)-th power of the radical of \(GF(2)[G],\) where \(G\) is an elementary abelian group of order \(2^m \) (see~\cite{B}). Self-dual RM-codes (i.e. some powers of the radical of the previously mentioned group algebra) exist only for odd \(m\).

The group algebra approach enables us to find a self-dual code for even \(m=2k \) in the radical of the previously mentioned group algebra with similarly good parameters as the self-dual RM codes.

In the group algebra
\[ GF(2)[G]\cong GF(2)[x_1,x_2,\dots, x_m]/(x_1^2-1,x_2^2-1, \dots x_m^2-1)\]
we construct self-dual binary \(C=[2^{2k},2^{2k-1},2^k]\) codes with property
\[{\rm RM}(k-1,2k) \subset C \subset {\rm RM}(k,2k)\] for an arbitrary integer \(k\).

In some cases these codes can be obtained as the direct product of two copies of \({\rm RM}(k-1,k)\)-codes. For \(k\geq 2\) the codes constructed are doubly even and for \(k=2\) we get two non-isomorphic \([16,8,4]\)-codes. If \(k>2\) we have some self-dual codes with good parameters which have not been described yet.


Reed--Muller code, Generalized Reed--Muller code, radical, self-dual code; group algebra; Jacobson radical

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bibitem{AK} Assmus, ~E.F. Key,~J.K., {it Polynomial codes and finite geometries,}

Chapter in Handbook of Coding Theory, edited by V. Pless and W. C. Huffman. Elsevier. (1995)

bibitem{B} Berman, ~S.D., {it On the theory of group code,} Kibernetika {bf 3} (1), 31--39, (1967)

bibitem{C} Charpin, ~P., {it Codes cycliques 'etendus et idA©aux principaux d'une alge`bre modulaire,} C.R. Acad. Sci. Paris, {bf 295} (1), 313--315, (1982)

bibitem{CL} Charpin, ~P, Levy-Dit-Vehel, ~F., {it On Self-Dual Affine-lnvariant Codes} Journal Combiunatorial Theory, Series A 67, 223--244, (1994)

bibitem{DL} Drensky, ~V., Lakatos, ~P., {it Monomial ideals, group algebras and error correcting codes,} Lecture Notes in Computer Science, Springer Verlag, {bf 357,} 181--188, (1989)

bibitem{J} Jennings, ~S. A., {it The structure of the group ring of a p-group over modular fields,} Trans Amer Math Soc {bf 50,} 175--185, (1941)

bibitem{KLP} Kasami, ~T. , Lin,~S, Peterson,~W.W., {it New generalisations of the Reed-Muller codes,}

IEEE Trans. Inform. Theory II-{bf 14,} 189--199, (1968)

bibitem{KYV} Kelarev, ~A. V.; Yearwood, ~J. L.; Vamplew, ~P. W.,

{it A polynomial ring construction for the classification of data.}

Bull. Aust. Math. Soc. 79 , {bf 2}, 213–-225, (2009)

bibitem{LM} Landrock, ~P., Manz, ~O., {it Classical codes as ideals in group algebras,} Designs, Codes and Cryptography, {bf 2,} Issue 3 273 -- 285, (1992)

bibitem{M}Muller, ~D.~E., {it Application of boolean algebra to switching circuit design and to error detection.} IRE Transactions on Electronic Computers, 3:6–-12, (1954)

bibitem{WS} MacWilliams,~F.J., Sloane, ~N.J.A., {it The Theory of Error-Correcting Codes,} North

Holland, Amsterdam, (1983)

bibitem{P} Pless, ~V., {it A classification of self-orthogonal codes over GF(2).} Discrete Mathematics {bf 3} 209--246, (1972)

bibitem{Wi} MacWilliams, ~F.J., {it Codes and Ideals in group algebras,} Univ. of North Carolina Press, (1969)


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