This paper presents a technique for increasing the reliability of arithmetic units. An error model is then presented; this model we'll represents the faulty behavior of many arithmetic units. The Residue Number System and its related properties are used in order to obtain a simple architecture (called Reliability Network, R-Net). The main characteristics of the presented technique are a significant reduction in the number of gales and a limited increase of global execution times. The extensive use of combinational logic makes it possible to implement the R-Net almost completely by means of Programmable Logic Arrays (PLA's). Finally, both the intrinsic regularity of the R-Net and its simple internal interconnection scheme make this approach suitable for a practical VLSI implementation.

A multiplier with multiple error correction capability

A method for performing continuous error detection and correction of arithmetic is disclosed which utilizes a redundant residue number system (RRNS) and signed values encoded using the method of complements. Prior art methods for error correction in an RRNS operate on positive magnitudes only. Correction of negative values encoded using the method of complements of the full range of the RRNS representation has not been previously solved. This paper demonstrates the prior art using an example error correction of a positive result and explains why negative numbers present a problem for error correction of arithmetic. Next, two new solutions for direct correction of negative numbers encoded using the method of complements is disclosed. In the first method, two groups of trial digit sets are tested; one group for positive numbers and the other for negative numbers. A basic proof for the method is detailed which uses the Chinese Remainder Theorem. In a second method also disclosed, negative values are directly corrected along with positive numbers using a single pass hardware pipeline. The second method is developed formally and is followed with an example of a direct error correction of a negative value. Finally, details of a pipelined error detection and correction circuit is provided. Because of the carry free properties of the RRNS number system, and because of the new correction methods, an error in any digit may be corrected with equal success, even after lengthy product summation operations. The new error correction method provides continuous correction of arithmetic since the circuit designs are fully pipelined and can deliver a corrected result every clock cycle. The new error correction method is used to provide error correction to signed arithmetic using a standard method of complements encoding; this enables flexible error correction of general-purpose RNS arithmetic for advanced applications such as convolution neural network (CNN) and matrix arithmetic.

Continuous Error Detection and Correction of Arithmetic in a Complement RRNS

This paper presents a new class of AN codes of high efficiency capable of correcting single errors in radix-r arithmetic with the binary coded digits—BCr—system. The errors corrected within a single radix-r digit are single errors of the binary digits with weight ±w i ≤ r–1,i=1,2,···, m, which are used to encode the BCr digits. The corresponding arithmetic unit is made out of slices, one for each BCr digit. The AN codes considered have a generator A of the form A = τ·p where τ, p odd and rp=n-1 ≤ τ < p. The paper establishes the selection criteria of r and p such that the code range of the AN codes is equal to M = rk± 1. The criteria are applied to the BCD system, and we determine all τ < 100 and p < 200 for the most important BCD codes with weights ± wi < r –1, i = 1, 2, 3, 4. For each BCD code, the paper gives the numbers τ < 100 for which AN codes exist, and the maximum code efficiency E = 2·m·k/(A-1) attained at some p < 200.

Error-Correcting Codes in Binary-Coded Radix-r Arithmetic

This paper considers AN arithmetic codes with radix r > 2 and binary-coded digits (BCr) using weights The error-correcting capability of the AN codes is single bit within any BCr digit, that is, the corrected errors are of the form ± wirj where wi are the weights of the BCr code The paper characterizes a class of AN codes having a generator of the form A = r · p where r|r − 1 or τ|r + 1 and p prime is greater than r − 1 or r + 1 It is shown that these codes, under certain conditions, are near perfect

Near-Perfect Codes for Binary-Coded Radix-r Arithmetic Units

Error-correcting coding schemes devised for binary arithmetic are not in general applicable to BCD arithmetic. In this paper, we investigate the new problem of using such coding schemes in BCD systems. We first discuss the general characteristics of arithmetic errors and define the arithmetic weight and distance in BCD systems. We show that the distance is a metric function. Number theory is used to construct a class of single-error-correcting codes for BCD arithmetic. It is shown that the generator of these codes possesses a very simple form and the structure of these codes can be analytically determined.

Chao-Kai Liu

Papers

Error-Correcting Codes in Binary-Coded-Decimal Arithmetic