We present a new class of asymptotically good, linear error-correcting codes These codes can be both encoded and decoded in linear time They can also be encoded by logarithmic-depth circuits of linear size and decoded by logarithmic depth circuits of size O(nlogn) We present both randomized and explicit constructions of these codes

Linear-time encodable and decodable error-correcting codes

The advent of digital VLSI technology demands a reappraisal of the most suitable algorithms for a given processing function. A cost function is developed for making quantitative comparisons between digital algorithm implementations including control and overheads. This provides a tool allowing different implementations of the same algorithm, and also different algorithms for the same function, to be compared. The cost function is chosen to characterise the algorithm and associated logic design independently of the circuit technology that might be used. In this way the techonology options can be introduced separately in designing a practical system. Also, configurations which achieve a higher throughput by using a proportionally larger quantity of hardware or higher logic speed are assessed as having the same effectiveness or cost. As an illustration, the costing is applied to FFT and Winograd Fourier-transform algorithms (WFTA). The results show the advantage gained by minimising the true costs including, particularly, control overheads, instead of just the number of arithmetic operations. Despite its fewer arithmetic operations, the WFTA is shown to be less efficient than the FFT except in the most fully parallel case.

Figures of merit for VLSI implementations of digital signal processing algorithms

We suggest a decoding algorithm of q-ary linear codes, which we call supercode decoding. It ensures the error probability that approaches the error probability of minimum-distance decoding as the length of the code grows. For n/spl rarr//spl infin/ the algorithm has the maximum-likelihood performance. The asymptotic complexity of supercode decoding is exponentially smaller than the complexity of all other methods known. The algorithm develops the ideas of covering-set decoding and split syndrome decoding.

On the complexity of minimum distance decoding of long linear codes

The computational work and the time required to decode with reliability E at code rate R on noisy channels are defined, and bounds on the size of these measures are developed. A number of ad hoc decoding procedures are ranked on the basis of the computational work they require.

The complexity of decoders--II: Computational work and decoding time

By concatenating linear-time codes with small, good codes, it is possible to construct in polynomial time a family of asymptotically good codes that approach the Shannon bound that can be encoded and decoded in linear time. Moreover, their probability of decoder error is exponentially small in the block length of the codes. In this survey, we will explain exactly what this statement means, how it is derived, and what problems in the complexity of error-correcting codes remain open. Along the way, we will survey some key developments in the complexity of error-correcting codes.

The Complexity of Error-Correcting Codes

Several classes of decoding rules are considered here including block decoding rules, tree decoding rules, and bounded-distance and minimum-distance decoding rules for binary parity-check codes. Under the assumption that these rules are implemented with combinational circuits and sequential machines constructed with AND gates, OR gates, INVERTERS, and binary memory cells, bounds are derived on their complexity. Complexity is measured by the number of logic elements and memory cells, and it is shown that minimum-distance and other decoders for parity-check codes can be realized with complexity proportional to the square of block length, although at the possible expense of a large decoding time. We examine tradeoffs between probability of error and complexity for the several classes of rules.

J. Savage

Papers

Complexity of decoders--I: Classes of decoding rules