The primitive idempotents of non-binary irreducible cyclic codes are difficult to construct. Berman [1967] gave the explicit expressions (without proof) of the primitive idempotents of irreducible cyclic codes of length pⁿ over GF(q); p an odd prime and n > 0; under the very strong assumption that q is a primitive root modulo pⁿ: Pruthi & Arora [1997] verified Berman's result. Arora, Batra, Cohen & Pruthi [2002] further obtained similar results for some very special cases. We obtained all the primitive idempotents of irreducible cyclic codes of length

• pⁿ; p an odd prime, p does not divide (q^f-1)/p if n > 1; here f is the multiplicative order of q modulo p (jointly with Bakshi, Dumir & Raka [2004]).
• 2pⁿ; p an odd prime, p does not divide (q^f-1)/p if n > 1; here f is the multiplicative order of q modulo p (jointly with Bakshi & Raka [2007]).
• 2ⁿ; q any odd prime power (jointly with Bakshi, Dumir & Raka [2008]).

In a subsequent paper, we proved that the generating idempotents of irreducible cyclic codes can be effectively evaluated, once they are known for irreducible cyclic codes of length pⁿ; p any prime (jointly with Bakshi & Raka [2008]). Thus we have almost completely solved the problem of determining the primitive idempotents of cyclic codes.

The problem of determining the weight distribution of a code is of great interest. It enables one to calculate the probability of undetected errors when the code is used purely for error detection and also gives us a measure of how good the code is at error correcting. However, computing the weight distribution of a code, even on a computer, can be a formidable problem. Many authors have worked on the problem of determining the weight distribution of irreducible cyclic codes using different techniques. We have obtained the weight distribution of

• all the irreducible cyclic codes of length 2ⁿ directly from their generating polynomials (jointly with Bakshi & Raka [2007]).
• some irreducible cyclic codes of odd prime power length (jointly with Bakshi [2011]).
• some binary reducible cyclic codes of odd prime power length (jointly with Bala [2011]).
• some q-ary reducible cyclic codes of odd prime power length (jointly with Bala [2012]).

Polyadic codes constitute a classical family of cyclic codes and are generalizations of quadratic residue codes, duadic codes, triadic codes, m-adic residue codes and split group codes, which have good error correcting properties (Leon, Masley & Pless [1984], Smid [1987], Pless & Rushnan [1988], Job [1992], Ding, Kohel & Ling[2000]). The necessary and sufficient conditions for the existence of polyadic codes of prime length was given by Brualdi & Pless [1989]. Ling & Xing [2004] gave the necessary and sufficient conditions for a certain subclass of polyadic codes. We have obtained the necessary and sufficient conditions for the existence of

• cyclic polyadic codes of prime power length and have also obtained some good codes arising from the family of cyclic polyadic codes (jointly with Bakshi & Raka [2007]).
• cyclic polyadic codes of arbitrary length in terms of the solvability of certain diophantine equations (jointly with Bakshi & Raka [2008]).
• Abelian polyadic codes of prime power length (jointly with Bakshi [2009]).

The results on the existence of polyadic codes, obtained by us, will enable one to construct polyadic codes of varying lengths and dimensions. Many interesting examples of good codes arise from the family of polyadic codes and the codes obtained from polyadic codes through minor modifications.

In a recent work, we have introduced some new weight enumerators of byte error-control codes and derived MacWilliams identities for them. We have also discussed some of their applications in modern computer systems using high density RAM chips.