Cyclic codes and quadratic residue codes over \(\mathbb Z_4\). (English) Zbl 0859.94018

Pless, Vera S.; Qian, Zhongqiang

IEEE Trans. Inf. Theory 42, No. 5, 1594-1600 (1996).

A set of \(n\)-tuples over \(\mathbb Z_4\) is a code over \(\mathbb Z_4\) if it is a \(\mathbb Z_4\) module. The authors show that any cyclic code \(C\) over \(\mathbb Z_4\) has generators of the form \((fh,2fg)\) where \(fgh=x^n-1\) over \(\mathbb Z_4\) and \(|C|=4^{\deg g} 2^{\deg h}\). The dual code \(C^\perp\) is shown to have generators of the form \((g^*h^*, 2f^*g^*)\) where \(f^*\) denotes the reciprocal polynomial of \(f\).
The authors define quadratic residue codes over \(\mathbb Z_4\). The extended quadratic residue codes over \(\mathbb Z_4\) of length 32 and 48 are studied in detail. The Gray map \((0\to 00\), \(1\to 01\), \(2\to 11\), \(3\to 10)\), of these two codes leads to binary nonlinear codes (of twice the length) that are shown to have higher minimum Hamming distances than the best known binary linear codes with comparable parameters.

Reviewer: T.Helleseth (Bergen)

MSC:

94B15	Cyclic codes
94B05	Linear codes (general theory)
94B40	Arithmetic codes

Keywords:

cyclic code; quadratic residue codes; Gray map