A coding theory approach to error control in redundant residue number systems. I. Theory and single error correction

Abstract

In this work, a coding theory approach to error control in redundant residue number systems is presented. We introduce the concepts of Hamming weight, minimum distance, weight distribution, and error detection and correction capabilities in redundant residue number systems. The necessary and sufficient conditions for the desired error control capability are derived from the minimum distance point of view. Closed-form expressions are derived for approximate weight distributions. Finally, computationally efficient procedures are described for correcting single errors. This paper is in two parts. In Part I, a coding theory framework is developed for redundant residue number systems, and an efficient numerical procedure is derived for a single error correction. Part I1 will deal with the extension of this framework and algorithms for correcting multiple errors and simultaneous error correction and detection.

Redundant residue number system based space-time block codes

“…The performance of systematic and nonsystematic RRNS codes and near-optimal decoding algorithms for soft decision decoding has been proposed in [37]. The similarities and advantages of RRNS over other non-binary codes like Reed-Solomon (RS) codes has been discussed in detail in [23], [25]- [27], [30]- [38].…”

“…Finally, the mapped symbols are constructed as space-time block codes and transmitted over multiple antennas. At the receiver, inverse operations including the use of Chinese Remainder Theorem (CRT) [30] to convert residues back to integers are implemented to recover the original information. We have derived upper bounds on the codeword and bit error probability of both non-systematic and systematic RRNS-STBC assuming ML detection and M-ary QAM constellation.…”

“…The CRT [28]- [30] is an important implementation of the reverse transformation from RNS and RRNS to the conventional weighted number system. According to the CRT, any u-tuple residue representation (r 1 , r 2 , ..., r u ), where 0 ≤ r i ≤ m i for i = 1, 2, ..., u, there exists one and only one integer N such that 0 ≤ N ≤ M r and r i = N (modm i ).…”

“…But recent advances in computing has enabled the use of RRNS as a channel code to improve error detection and correction [23], [27], [30]- [38].…”

A signed digit adder with error correction and graceful degradation capabilities

“…The self-checking capabilities of adders with carry free implementations are widely known, while there are not many works proposing adders which provide also error correction capabilities. The most widely applied techniques to obtain error correction in adder circuits are based on time-redundancy [7] or on the residue number system representation [8], [9]. The goal of this paper is to introduce novel fault localization and graceful degradation methodology to be applied to a self checking adder [10] which uses the signed digit representation [11].…”