Research On Some Problems Of Residue Codes Over Finite Rings

The paper investigates biquadratic residue codes over the finite rings Z4 and quadratic residue codes over Fl+vFl+v2Fl(where v3=v and l is a prime) and Z25+uZ25(where(u2=u)). The details are given as follows:(1) The first part, the notion of eight biquadratic residue codes of length p over Z4 is defined and their idempotent generators are given. Then, the relations between the eight codes are researched and some properties of the codes are discussed. The generating polynomials of their dual codes are calculated. Finally, extended biquadratic residue codes over Z4 and their properties are studied.(2) The second part, the notion of four quadratic residue codes over the ring Fl+vFl+v2Fl are defined in terms of their idempotent generators, where v3=v and l is an odd prime. The relations between the four codes are discussed. Then, the relations between the four codes and their dual codes are obtained.(3) The third part, the structure over the ring Z25+uZ25 are studied, where u2=u. The ontion of the quadratic residue codes over the ring Z25+uZ25 are defined in terms of their idempotent generators. The relations between the quadratic codes are researched and some properties of the codes are discussed. Furthermore, extended quadratic residue codes over Z25+uZ25 and their properties are studied.