Related Study On T(=1,2)-designs Based On Linear Codes

Designs as an important branch of combinatorics,which can be widely used in coding theory,cryptography,communication and statistics.And linear codes with a few weights as a significant part of coding theory,which also can be widely applied in secret sharing schemes,authentication codes,association schemes,data storage systems and combinatorics.It is well known that there is a very close relationship between coding theory and design.On one hand,a linear codes over any finite field can be derived from the incidence matrix of a t-designs.On the other hand,the supports of all codewords with a fixed weight in a linear or nonlinear codes might hold a t-design under certain conditions.In this paper,some 1-designs and 2-designs are constructed based on Assmus-Mattson theorem and affine invariant codes theory,the parameters of the designs are determined,and we illustrate the validity of our results using Magma program.The main work is as follows:(1)We determine the parameters of infinite families of 2-designs based on t-wo classes of linear codes.One is based on ternary linear codes.Another is based on the narrow-sense primitive BCH codes C(q,qm-1,δ3,1),and then we show that C(q,qm-1,δ3,1)⊥ also hold 2-designs.(2)We select four-valued Walsh spectrums Boolean functions and three-weight linear codes to construct four classes of linear codes with a few weights,and their parameters and weight distributions are determined.Meanwhile,we calculate the parameters and weight distributions of their dual codes.Then,we study infinite families of 1-designs from one kind of codes.(3)We select five-weight binary linear codes holding 2-designs to construct a class of linear codes with a few weights,and their parameters and weight distribu-tions are determined.Then,we derive infinite families of 1-designs from them.