Error Detection with Real-Number Codes Based on Random Matrices

Abstract

Some well-known real-number codes are DFT codes. Since these codes are cyclic, they can be used to correct erasures (errors at known positions) and detect errors, using the locator polynomial via the syndrome, with efficient algorithms. The stability of such codes are, however, very poor for burst error patterns. In such conditions, the stability of the system of equations to be solved is very poor. This amplifies the rounding errors inherent to the real number field. In order to improve the stability of real-number error-correcting codes, other types of coding matrices were considered, namely random orthogonal matrices. These type of codes have proven to be very stable, when compared to DFT codes. However, the problem of detecting errors (when the positions of these errors are not known) with random codes was not addressed. Such codes do not possess any specific structure which could be exploited to create an efficient algorithm. In this paper, we present an efficient method to locate errors with codes based on random orthogonal matrices.