Irreducible Cyclic Codes and Polyadic Codes: Generating Idempotents, Weight Distributions, Existence Conditions

In this book, explicit expressions of all thegenerating idempotents of irreducible cyclic codes oflength p^n and 2p^n over GF(q) are obtained, where pis an oddprime, p does not divide (q^f-1)/p if n > 1 and f isthe multiplicative order of q modulo p. Also thegenerating idempotents of all theirreducible cyclic codes of length 2^n over GF(q),where q is any odd prime power, are determined.It is also proved that the generating idempotents ofirreducible cyclic codes can be effectivelyevaluated, once they are known for irreducible cycliccodes of length prime power length. The weightdistributions of all the irreducible cyclic codes oflength 2^n are also obtained directly from theirgenerating polynomials. The necessary and sufficientconditions for the existence of cyclic polyadic codesof prime powerlength and of arbitrary length are obtained. The results on the existence of polyadiccodes, obtained by us, will enable one to constructpolyadic codes of varying lengths and dimensions.Many interesting examples of good codes arise fromthe family of polyadic codes.