Coding Theory Information Theory and Radar

摘要

The discrete Heisenberg-Weyl group provides a unifying framework for a number of important sequences significant in the construction of phase coded radar waveforms, in communications as spreading sequences, and in the theory of error correcting codes. Among the sequences which can be associated with the Heisenberg-Weyl group are the first and second order Reed-Muller codes, the Golay or Welti sequences, and the Kerdock and Preparata codes , which are non- linear binary error correcting codes containing more codewords for a given minimum distance than any linear code. The Kerdock codes are associated with decomposition of the Heisenberg-Weyl group into disjoint maximally commutative subgroups. It is a surprising fact that a certain general class of Golay sequences exist within the Kerdock codes. This had had previously been noted by Davis and Jedwab. Golay sequences are pairs of sequences of unimodular complex numbers with the property that the sum of their individual auto- correlation functions forms delta spike or thumb tack. These sequences have found application in the construction of radar waveforms and in modulation schemes for communications.