Fourier transform over finite groups for error detection and error correction in computation channels

Abstract

We consider the methods of error detection and correction in devices and programs calculating functions f: G → K where G is a finite group and K is a field. For error detection and correction we use linear checks generated by convolutions in the field K of the original function f and some checking idempotent function δ : G → , 1 For the construction of the optimal checking function δ we use methods of harmonic analysis over the group G in the field K . Since these methods will be the main tools for the construction of optimal checks, we consider the algorithms for the fast computation of Fourier Transforms over the group G in the field K . We solve the problem of error detecting and correcting capability for our methods for two important classes of decoding procedures (memoryless and memory-aided decoding) and consider the question of syndrome computation for these methods. We describe also properties of error correcting codes generated by convolution checks.