PhDThis 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.
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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
