Group structure of Hadamard memories

The hookup of Hadamard memories is expressed in terms of the structure of a commutative group over the index set {1,2,. . .N}, where N is the data bit length, assumed to be a power of 2. The group relations occur in the form of triads of indices. A quadratic memory, hooked up according to the triad group structure, turns out to be a Hadamard memory. The latter is an associative memory that has Hadamard vectors as stable states; the memory has perfect associative recall. For data bit lengths that are special powers of 2, the triads can be restricted to be shift invariant, and the memory hookup is then specified by a single integer, the seed. For those cases, a VLSI implementation is possible which places all dendrite lines in a single plane. The relation between the triad group and Hadamard vectors is mentioned. In the simplest continuum model of the Hadamard memory, the dynamics of the neural net is found to be invariant under the triad group