Applications Of Character Matrix In Coding Theory

This paper uses the character matrix of a finite group to give some results of algebraic coding theory.Details are as follows:Minimum Hamming distance of irreducible cyclic code.Suppose that l1l2|(q-1),where l1,l2 are distinct primes.Then,using the character matrix of a finite group,we give all primitive idempotent elements of the ring Fq[x]<xl1m1l2m2-1>,and determine minimum Hamming distances of irreducible cyclic codes of length l1m1/l2m2.Suppose that p|(q-1)and q ≡ 3(mod 8),where p is odd prime.Then we give all idempotent elements of the rings Fq[x]/<x4pn-1>and Fq[x]/<x8pn-1>,and determine the minimum Hamming distances of irreducible cyclic codes of length 4pn and 8pn.Weight distribution of cyclic code.Suppose that Fr*＝<α>,where r = qm1m2 and gcd(m1,m2)=d.Let α1=αr-1/qm1-1 and α2=αr-1/qm2-1,then Fqm1=<α1>and Fqm2*=<α2>Suppose that gcd(e,qm2-1/qd-1).When d=1,2,we use Gauss sum to give the weight distribution of a cyclic code C={c(a,b):a∈Fqm1,b∈Fqm2},c(a,b)=(T1(aα1 i)+T2(bα2ei))i=0n-1,where Ti are the trace functions from Fqamt to Fq,i=1,2.Moreover,when d=1,we prove that C is a optimal code which attains Griesmer bound.It generalizes results in[33,34].Weight distribution of consta-cyclic code.Let an=λ and 0≠λ∈Fq.We obtain the weightdistribution of A-consta-cyclic code C of length nm with parity-check polynomial h(x)=(xm-aξst)(xm-aξs(t+1)...(xm-aξs(t+r-1),where ξ is a n-th primitive root in Fq.We also give weight distributions of λ-consta-cyclic codes of length nm explicitly in several cases:(1)r=1、n>1;(2)r=2,m = 2,n>2;(3)r = 2,m = 3,n>3;(4)r = 3,m = 2,n>4.Minimum Hamming distance of character code.Using character matrix,we construct a new character code Cq(0,1,...,r-1,n-r+1,...,n;n),determine its minimum hamming distance 2r,and it has parameters[2n,2n-2Sn(r-1),2r];moreover,we use the new group character code to construct a family of unit memory convolutional codes with parameters[2n,2n-Sn(r-1),Sn(r-1);1,2r].MDS convolutional code and quantum MDS convolutional code.Let q>5 and q≡±5(mod 13).We use consta-cyclic code of length over to construct new MDS convolutional code with parameters where we investigate consta-cyclic code with dual-containing,construct self-orthogonal convolutional code,and obtain two classes of new quantum MDS convolutional code.