The Constructions Of Cyclic Codes With Definite Minimum Distances.

We have three parts in this paper:The first part: We introduce some known results about residue codes and BCH codes,and state the main contributions in this thesis.The second part:We give some preliminaries; mainly including some concepts about cyclic codes, primitive roots and the method of construction for residue codes.The third part: Firstly we extend the length of residue codes,and then construct a family of cyclic codes with definite parameters by analyzing the characters of residue codes and using the primintive roots of modulo p^k and modulo 2_p^k,where p is an odd prime. Secondly,under the inspiration of literature,we construct another family of cyclic codes with definite lengths,definite dimension,and definite minimum distances by using the character of the codes's length. The following statements are the main results:Theorem 3.1.1:let p be an odd prime, q be a prime number, ( p , q ) = 1,αbe a primitive (p^k)^th root of unity over a extended field of F_q ,letβbe a primitive foot module p k, suppose g(x)=(?),then(3)the parameters of the cyclic codes generated by g(x)is[p^k,p^k-1,p] over F_q.Theorem 3.2.1:let q ,p be odd primes,(p,q)=1,γbe a primitive (2p^k)^th root of unity over a extended field of F^q , letα=γ², thenαis a primitive (p^k)^th root of unity,supposeβis a primitive root module 2p^k, g(x)=(?),then (3)the parameters of the cyclic codes generated by g(x) is [2p^k,p^k-1,d≤p] over F_q .Theorem 3.3.1:let q be a prime power, and q≥3,C be a cyclic code with the length n=q^t+1, whose generator polynomial is g(x)=m₁(x),then the parameters of C is [q^t+1,q^t+1-2t,d],where d=2,3 or 4.Theorem 3.3.4:let q be a prime power, and q≥4, C be a cyclic code with the length n=q^t+1 and generator polynomial g(x)=m_o(x)m₁(x), then the parameters of C is [q^t+1,q^t-2t,4].Theorem 3.3.5:the parameters of the dual code C^⊥of C is [q^t+1,2t+1,d'],d'≥q^t-2q^t-1+1 where C is generated by g(x)=m_o(x)m₁(x) with length n in theorem 3.3.2.