Generalized TCH codes via Zech logs

TCH codes are a class of binary nonlinear cyclic codes that have been used in several systems (e.g. mobile satellite, Ultra-Wideband) for tasks like coding and synchronization. The original generating method of this class restricts the length (or period) of the codewords to the set f2;4;16;256;65536g, although other power-of-two lengths are possible by search and trial ad-hoc methods. In this paper we propose an alternative generation method based on the algebraic properties of Zech logarithms that allow us to generalize the construction in a systematic manner. This new procedure is less complex and the results are comparatively better in terms of codeword length choices, code size and minimum distance.