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.

翻译：我们将Cherubini与Micheli提出的阴影码推广至包含任意次数的基本多项式，并证明将基本多项式限制为一次或更低次数可优化码的参数特性。所得码仅在极长码块长度（例如$\geq 2^{20}$）条件下优于著名的Delsarte-Goethals码，因此可能仅适用于极高噪声信道的实际场景。