Bit-level erasure decoding beyond design distance ofReed-Solomon codes over GF(2m)

This paper explores the question of how to extend the capability of a Reed-Solomon code beyond the design distance in the case that symbols are affected on the bit level as would occur, for example, if data is obtained via packet interleaving on bits of the information sequence, so that packet losses result in bit-level erasures. It is shown that it is possible to decode beyond the erasure design distance for many codes. On the basis of simulation, the correction capability and plots showing the probability of uncorrectable erasures are provided. Extended summary Background: In many network-based multimedia applications, latency is a critical performance factor. In the interest of controlling latency, many applications avoid potential retransmission inherent in the guaranteed delivery of TCP protocol by employing UDP protocols as a base layer and protecting the data with a combination of forward error correction, employing packet retransmission only when necessary. In such cases, data may be interleaved so that the loss of a data packet results in only a small loss of a codeword, which may be reconstructed using an error correction code. Since a lost packet is known to be lost, this results in erasures in the data, as opposed to errors, essentially doubling the correction capability of the code. This paper explores the question of how to extend the capability of a Reed-Solomon code beyond the design distance in the case that symbols are affected on the bit level as would occur, for example, if packet interleaving was performed on bits of the information sequence, so that packet losses result in bit-level erasures. It is shown that it is possible to decode beyond the design distance for many codes. On the basis of simulation, the correction capability and plots showing the probability of uncorrectable erasures are provided. For example, for a (255,250) RS code, the method is able to correct 19 bit erasures more than would be expected using conventional bounded distance decoding. There has recently been considerable work done on decoding beyond the design distance. Sudan [1] then GuruswamiSudan [2] published work on decoding beyond the d/2 bound. The algorithm was put in a computationally efficient form by Kötter [3] and Roth-Ruckenstein [4]. These works rely on some algebraic geometry. The method of this paper, on the other hand, is based on conventional syndrome decoding. While not providing necessarily the same capability as the Guruswami-Sudan method (for example, it deals only with erasures and not errors), the method of this paper is quite straightforward to implement and does not require a background in algebraic geometry. In the case that the number of erasures exceeds the design distance there are an insufficient number of conventional syndrome equations, so that additional syndrome equations must be created to solve for more erasures. These are obtained by exploiting the fact that (x + y) = x + y for x, y ∈ GF (2). If X0b = s0 is a matrix equation representing the error-value problem (for erasure correction), then it is shown that additional