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Cyclic codes and quadratic residue codes over Z4

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2 Author(s)
V. S. Pless ; Dept. of Math. Stat. & Comput. Sci., Illinois Univ., Chicago, IL, USA ; Zhongqiang Qian

A set of n-tuples over Z4 is called a code over Z4 or a Z4 code if it is a Z4 module. We prove that any Z4-cyclic code C has generators of the form (fh, 2fg) where fgh=xn-1 over Z4 and |C|=4deg g 2deg h. We also show that C has generators of the form (g*h*, 2f*g*). We show that idempotent generators exist for certain cyclic codes. A particularly interesting family of Z4 -cyclic codes are quadratic residue codes. We define such codes in terms of their idempotent generators and show that these codes also have many good properties which are analogous in many respects to properties of quadratic residue codes over a field. We show that the nonlinear binary images of the extended QR codes of lengths 32 and 48 have higher minimum weights than comparable known linear codes

Published in:

IEEE Transactions on Information Theory  (Volume:42 ,  Issue: 5 )