Constructions Of Good Codes And Asymptotically Good Codes Over Finite Fields

Optimal and asymptotically good codes,approximately symmetric informationally complete positive operator-valued measures(ASIC-POVM),and permutation polynomials over finite fields have many important applications in communication and cryptography.In this paper,we investigate q-ary quantum codes,asymptotically good quasi-cyclic codes of fractional index,ASIC-POVM,and permutation trinomials over finite fields.Firstly,through the analysis of q²-cyclotomic cosets modulon,a sufficient and necessary condition of whether a cyclic code is Hermitian dual-containing is presented.Then several classes of quantum MDS codes are obtained by Hermitian construction.In addition,we get some numerical results on cyclic codes,as well as a recursive construction of Hermitian dual-containing cyclic codes from a known Hermitian dual-containing cyclic code.Secondly,we study a more general class of quasi-cyclic codes of fractional index in the module R₍km × R₍lm=F_q[X]/<X^km-1>× Fq[X]/<X^lm-1>.The parity check polynomials and encoders of this class of quasi-cyclic codes are obtained.The encoder enables us to establish a relationship between quasi-cyclic codes of fractional index in Rkm×Rlm and those of index two in Rm×Rm,while this relationship admits asymptotically good quasi-cyclic codes of fractional index.Next,with the help of character sums of two special kinds of functions over finite fields,i.e.,additive character sum of PN functions and mixed character sum of PF functions,two classes of ASIC-POVMs are constructed.Firstly the accurate value of character sums are calculated,and then POVM elements are defined.For the second construction,we show that the PF functions possess recursive constructions,which ensures recursive constructions of ASIC-POVMs,giving some classes of infinite families of ASIC-POVMs.Finally,we investigate the construction of permutation trinomials over finite fields.By rewriting the trinomials into fractional polynomials,the problem is changed to prove the fractional polynomials are permutation of ?d,the d-th root of unity in the finite fields.We cut ?d into two parts and prove that the fractional polynomials are piecewise permutations of ?d.Notably,we show that one can use self-reciprocal polynomials to get a recursive construction of permutation polynomials.