Quaternary Cyclic Codes
Abstract
Cyclic codes are considered for the quaternary alphabet, the field K = GF(2) squared. If A is a (k,n) (n odd) quaternary group codes - i.e., a k- dimensional subspace of ordered n-tuples of K elements - then A is isomorphic via the Solomon-Mattson polynomials, to a subgroup of the direct product of K with r copies of L. (L is the smallest field over K containing the nth roots of unity and r is the number of irreducible factors of x to the n power + 1/x + 1 over K.) Let d(A,K) be the minimum weight of non-zero vectors of A. For p, a prime, and A, a (k,p) cyclic K code, d(A, K) greater than or equal to d(A,F) where d(A,F) is the Bose-Chaudhuri bound for the corresponding binary cyclic codes of the same order (if there is one). Number theoretic methods are introduced to improve the Zierler-Gorenstein lower bound for certain primes p. For p such that 2 has multiplicative order p- 1, there exists (p +1/2, p) cyclic codes with d(p) greater than or equal to 3 is 3 is not a quadratic residue of p, d(p) greater than or equal to 4 if 3 is a quadratic residue of p, and d greater than or equal to 5 if both 3 and 5 are quadratic residues of p. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Sep 13, 1961
- Accession Number
- AD0263608
Entities
People
- Gusatave Solomon
Organizations
- Massachusetts Institute of Technology