Research On The Construction Of Linear Block Codes Over Finite Fields

Using normal elements and circular permutations, this dissertation gives the method of the construction of linear codes with the best parameters; then it analyses the construction of linear codes over the constructed finite fields which can be used for coding for two-dimensional signal spaces. The main results are as follows: We generalize the idea of constructing codes over a finite field F_q by evaluating a certain collection of polynomials at elements of an extension field of F_q. The method we employ for extensions of arbitrary degrees is different from the approach in [9]. We make use of a normal element and circular permutations to construct polynomials over the intermediate extension field between F_q and F_q^t denoted by F_q^s where s divides t. The polynomials we constructed are F_q-linearly independent and return elements in F_q when they are evaluated at elements of the extension fields. We give the value of the length n, the dimension k and the lower bound of the minimum distance d of the new linear codes. It turns out that many codes with the best parameters can be obtained through our construction and improve the parameters of Brouwer's table. Some codes we get are optimal by the Griesmer bound. In this paper, new classes of linear block codes over finite fields of the algebraicinteger ring of quadratic number fields Q(d^1/2) modulo irreducible elements with norm p or p² are presented. These codes can correct one error which takes from the cyclic subgroup of the multiplicative group of the finite fields. The results presented in this paper extend the corresponding results of previous papers.