Trace Expressions Of Constacyclic Codes Over Finite Fields And Cyclic Codes Over Z_p²

We have three parts in this thesis.The first part:We give preliminary knowledge about codes,review the known results in constacycli ccodes and cyclic codes over the ring Z_p² ,and state the main contributions in this thesis.The second part:Firstly,we give the trace expression ofλ-constacycliccode over finite field F_q .Then,we give the trace expression of irreducible Negacylic code and its index. Finally,we provide some examples.By these examples we can see that the trace expression of a cyclic code is very important in getting the weight distribution of the code and understanding the construction of the cyclic code.Theorem 2.3.1:If C is aλ-constacycliccode over the finite field F_q , g (x)∈F_q is its generator polynomial and h (x)= (xⁿ-λ)g(x) is its check polynomial,let |λ|=m, x^mn-1=(xⁿ-λ)h₁ (x)=g(x)h(x)h1(x)h₁(x)h₁ (x)= g(x)p₁(x)p₂(x)…p_s(x) p₁(x),p₂(x)…p_s(x)are pairwise different irreducible polynomials of the degree di (1≤i≤s), then for every codeword c = ( c₀ ,c₁,…c_n-1 )∈C,there existβ_i∈F_q^di)(1≤i≤s) satisfying c_λ=sum from i=1 to S( T_i)(β_iα_i^-λ,whereαis a root of h (x), T_i is a trace map from F_q^di to F_q .Theorem 2.3.4:If C is a irreducible Negacylic code over the finite field Fq with the check polynomial h ( x)∈Fq of the degree k ,then for every codeword c = ( c₀ ,c₁,…c_n-1 )∈C there existsβ∈F_q^k satisfying c_λ= T(βα^-λ)(0≤λ≤n-1),whereαis a root of h (x), Ti is a trace map from F_q^k to F_q .Theorem 2.3.5:If C is a irreducible Negacylic code over the finite field Fq with the check polynomial h (x)∈F_q which is a primitive irreducible polynomial of the degree k ,then C is a equiweight cyclic code with index [(q^k-1)/2,k,(q^k-1(q-1))/2].The third part:We give the trace expression of a cyclic code over the residue class ring Z_p² Theorem 3.3.1:Let p be a prime and put q = p², x^pm =g(x)h(x),where h(x) is a primitive basic irreducible polynomial in Z_q.If C is a cyclic code over Z_q, then C =...