On Binary Shadow Codes

We generalize the shadow codes of Cherubini and Micheli to include basic polynomials having arbitrary degree, and show that restricting basic polynomials to have degree one or less can result in improved code parameters. The resulting codes improve upon the well-known Delsarte-Goethals codes only in the regime of extremely long block lengths ($\geq 2^{20}$, say), and so are likely to be of practical interest only for very noisy channels.