Q-codes

Creators: Pless, Vera

Abstract

We introduce a new infinite family of quaternary cyclic (n,(n+1)2) and (n,(n − 1)2) codes which include Q. R. codes when n is prime. These codes are defined in terms of their idempotent generators and exist for all odd n. Every self-dual (s.d.) or strictly self-dual (s.s.d.) extended cyclic code is an extended Q-code. As for Q. R. codes, there is a square root bound on the odd-like minimum weight. We can tell by the factors of n whether s.d. or s.s.d. extended cyclic codes exist and when all or some extended Q-codes are s.d. or s.s.d. We know when Q-codes have binary idempotents and when the binary subcode is just the all-one vector. A table of Q-codes of modest lengths is given. Minimum weights, idempotents and duals are identified.

Additional Information

© 1986 Published by Elsevier. Received 28 March 1986. This work was supported in part by the National Science Foundation Grants MCS-8201311, R11-8503096 and NSA Grant MDA 904-85-H-0016. I wish to acknowledge very helpful conversations with Dr. John Masley. In particular, he proved major portions of Theorems 8, 10, and Corollary 2 of Theorem 11. I wish to thank Professor Richard Wilson for pointing out that Theorem 20 is part of Mann's theorem and for other helpful comments. I wish to thank Professor Noburu Ito for many helpful comments and for a careful reading of this paper leading to several corrections. The minimum weights, weight distributions, and groups of the codes in Tables I and II were computed by the CAMAC system [2] running on the University of Illinois at Chicago computer.

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Q-codes

Abstract

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