Generating Idempotents Of Quadratic Residue Codes Over Finite Fields

We have five parts in this thesis.In the first part we review some known results about the generating idempotents of quadratic residue codes over the finite fields, and introduce the main contributions in this thesis.In the second part we give some preliminaries about codes ,including some concepts about linear codes ,cyclic codes, quadratic residue codes and generating idempotents.In the third part we obtain expressions of generating idempotents of residue codes over the finite fields F₂ , F₄ ,F₂^t .In the fourth part we obtain expressions of generating idempotents of quadratic residue codes over the finite field F₃ , F₉ ,F₃^t .Finally in the fifth part expressions of the generating idempotents of quadratic residue codes over the finite field F₅ , F₂ 5,F_5t are given.This main results in the thesis are as follows: Theorem1.The four generating idempotents of quadratic residue codes on the finite field F5 are(?)Where Q pis a set of quadratic residues modulo p and N p is a set of nonquadratic residues modulo p .αis a primitive root of unity of irreducible polynomial x 2 ? x+2over the field F5 .Theorem2.The four generating idempotents of quadratic residue codes on the finite field F2 5are (?)Where Q pis a set of quadratic residues modulo p and N p is a set of nonquadratic residues modulo p .αis a primitive root of unity of irreducible polynomial x 2 ? x+2over the finite field F5 .