Coding theory and discrete transforms
We investigate a new approach to the construction of linear block codes, the so-called loop transversal approach. First, we use a greedy algorithm to construct syndrome functions of binary lexicodes up to high channel lengths, presenting tables of dimensions for these codes. The graphs of the syndrome functions turn out to have curious fractal properties. In order to investigate these functions, we then consider them as polynomials in subfields of the quadratic closure of GF(2). Passing from such a polynomial function to its coefficient sequence provides a linear transform, analogous to the discrete Fourier transform. Although the matrices of the transforms are non-sparse and increase exponentially in size, we are still able to invert them explicitly. The inverse transform matrices again have a fractal structure, including the Sierpinski triangle in their first row. The transforms of the syndrome functions are exhibited. Two features of these transforms are readily apparent. The first is the apparent simplicity of the transforms when compared with the original syndromes. The second is the similarity of the transforms for the various syndromes. This similarity illustrates the way in which the syndrome functions generalize the logarithm function;Secondly, we use the transform to analyze certain spaces of natural number functions that include the syndromes of the codes. Transforms of functions in these spaces exhibit a martingale property;Finally, we present an alternative, more elementary approach (compared with (Le2)) to the solution of a problem posed by J. H. Conway on exponentiation in the quadratic closure of GF(2).