Construction Of Optimal Codes And Research On Properties Of Cyclic Codes

In this thesis,we mainly study the weight distribution of cyclic codes over finite fields,the construction of few-weight codes over finite rings,and the algebraic structure of polynomial cyclic codes and their minimum distances.The details are as follows:1.The trace codes over the finite chain ring R1 and the finite non-chain ring R2 are studied.Using simplicial complexes(or down-sets)to construct the defining sets and character sums,we give the weight distribution of the trace codes over different rings.A linear Gray map yields three infinite families of few-weight codes as follows:?The Gray image ?(CL)of the codes CL over R1=F2+uF2+u2F2 are binary three-weight codes;?We construct two Gray images over R2=Fp+uFp which are ?(CL1)(p-ary two-weight codes)and?(CL2)(p-ary three-weight codes).In particular,we give conditions which all image codes reach the Griesmer bound.Furthermore,we verify codewords of the Gray image constructed are minimal.Comparing a large number of references,the above two kinds of three-weight codes are new codes with good parameters.2.Upper and lower bounds on the largest number of weights in a cyclic code of given length,dimension and alphabet are given.An application to irreducible cyclic codes is considered.Sharper upper bounds are given for the special cyclic codes(called here strongly cyclic),whose nonzero codewords have period equal to the length of the code.Lower bounds appear weak so far,being mostly linear in k,when the upper bounds are exponential.The Melas and Zetterberg codes provided lower bounds exponential in the dimension and it is worth extending these bounds to other range of parameters.The results on the weights of q-ary Reed-Muller and Hamming codes,while of interest in their own right only provide lower bounds that are polynomial in the dimension.3.The polycyclic codes are described from the linear transformation.Both linear codes and cyclic codes are linear subspaces in terms of algebraic structure.The authors studied constacyclic codes by the theory of invariant subspaces from linear algebra[59].Polycyclic codes are a powerful generalization of cyclic and constacyclic codes.Their algebraic structure are also studied here.As an application,a bound on the minimum distance of these codes is derived which outperforms,in some cases,the natural analogue of the BCH bound.