Z4-codes and their gray map images as orthogonal arrays and t-designs

Abstract

This thesis discusses various connections between codes over rings, in par- ticular linear Z4-codes and their Gray map images as orthogonal arrays and t-designs. It also introduces the connections between VC-dimension of binary codes and the strengths of the codes as orthogonal arrays. The second chapter concerns codes over rings. It is known that if we have a matrix A over a eld F, whose rows form a linear code, such that any t columns of A are linearly independent then A is an orthogonal array of strength t. I shall begin with generalising this theorem to any nite commutative ring R with identity. The case R = Z4 is particularly important, because of the Gray map, an isometry from Zn 4 (with Lee weight) to Z2n 2 (with Hamming weight). I determine further connections that exist between the strength of a linear code C over Z4 as an orthogonal array, the strength of its Gray map image as an orthogonal array and the minimum Hamming and Lee weights of its dual C?. I also nd that the strength of a binary code as an orthogonal array is less than or equal to its strong VC-dimension. The equality holds for linear binary codes. Furthermore, the lower bound is also determined for the strength of the Gray map image of any linear Z4-code. 4 Moreover, I show that if a code over any alphabet is an orthogonal array with a certain constraint then the supports of the codewords of some Hamming weight form a t-design. Furthermore, I prove that if a linear Z2- code satis es the t-mixture condition, then such a code is an orthogonal array of strength t. I then investigate if such connection also exists for non- linear Gray map images of linear Z4-codes, and prove that it does for some values t.