Quadratic Functions And Its Application In Codes And Combinatorial Designs

Combinatorial t-designs have been an interesting topic in combinatorics for decades.They have nice applications in coding theory,finite geometries and engineering areas.We know that t-designs can be constructed from image sets of a fixed size of some special polynomials.Till now only a small amount of work on constructing t-designs from special polynomials has been done,and it is in general hard to determine their parameters.This paper constructst-designs from the quadratic polynomial x⁴+x³ over F_2ⁿ and determine their parameters.We yield 2-(2ⁿ,3·2^n-2,2^n-2(3·2^n-2-1))designs for n even and 3-(2ⁿ,2^n-1,2^n-1(2^n-2-1))designs for nodd.BCH codes with efficient encoding and decoding algorithms have many applications in communication,cryptography and combinatorial design.This paper studies a class of linear codes of length (q^m-1)/2 over Fq with special trace representation,where q is an odd prime power.With the help of the inner distribution of some subsets of association schemes of quadratic forms,we determine the weight enumerators of these codes.Determining some cyclotomic coset leaders ?_i of cyclotomic cosets modulo (q^m-1)/2,we prove that narrow-sense BCH codes of length (q^m-1)/2 with designed distance (?) have the corresponding trace representation,and have the minimal distance d=?_i and the Bose distance d_B-?_i,where(?).