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On r-fold Complete Weight Enumerator of r Codes
by
Irfan Siap
The Ohio State University
Coauthors: Dijen K. Ray-Chaudhuri
MacWilliam's identity which relates the Hamming weight enumerators of a code and its dual has played an important role in Coding Theory. Researchers have considered several generalizations of Hamming weight enumerators for instance complete weight enumerator, Lee weight enumerator etc and proved MacWilliam's identities for these instances. In our work we have introduced an r-fold complete weight enumerator of r linear codes of same length and over the same finite field of q elements. We have established a relation (MacWilliam's type identity) between r-fold complete weight enumerator of r linear codes and their duals. Also we have similarly introduced Hamming and Lee r-fold weight enumerators of r linear codes and MacWilliam's type identities are obtained as corollaries. Moreover, from our theorem all other known MacWilliam's identities follow immediately as corollaries. Especially, MacWilliam's identity for r-ply weight enumerator of a linear code introduced by K. Shiromoto (1996) follows as a corollary to our main theorem by specializing the variables. We have determined the ring of invariants of 2-ply weight enumerator of binary self dual linear codes and binary self dual codes with lenghts divisible by 8 by applying invariant theory.
Date received: January 23, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-56.