Bounded-angle iterative decoding is a modified version of conventional iterative decoding, conceived as a means of reducing undetected- error rates for short low-density parity-check (LDPC) codes. For a given code, bounded- angle iterative decoding can be implemented by means of a simple modification of the decoder algorithm, without redesigning the code.

Iterative decoders for well-designed LDPC codes are inherently capable of detecting as well as correcting errors. In a given instance, an error is detected if a code word is not found at the conclusion of iterations: that is, an error is detected if the estimated code symbols at the end of iterations do not constitute a valid code word. However, when this approach is followed without modification, the maximum undetectederror rates obtained for short LDPC codes are typically too high for many applications.

Bounded-angle iterative decoding is based on a representation of received words and code words as vectors in an n-dimensional Euclidean space (where n is an integer). In bounded-angle iterative decoding as in conventional iterative decoding, the estimates of the decoder are rejected (an error is detected) if a code word is not found before the maximum allowed number of iterations. Conversely, iterations may be stopped as soon as a code word is detected. The difference between bounded-angle and conventional iterative decoding manifests itself once a code word has been detected. At this point, the decoding algorithm computes the angle in the n-dimensional Euclidean space between the received word and the detected code word. If this angle is less than an arbitrarily specified threshold angle, θd, then the detected code word is accepted; if this angle exceeds θd, then the detected code word is rejected.

The undetected-error rate in bounded-angle iterative decoding is necessarily less than that in conventional iterative decoding because the modified decoding algorithm rejects at least the same code words as does the conventional decoding algorithm. By rejecting more code words, the modified algorithm reduces the undetected-error rate while increasing the overall error rate. Through judicious choice of θd, optimized as a function of signal-to-noise ratio Eb/N0 (see figure), one can reduce the decoder’s maximum undetected- error rate by orders of magnitude while increasing its overall error rate by amounts that are negligible for all values of Eb/N0 high enough to have produced a low error rate using the unmodified decoder. The main value of the modified algorithm lies in the possibility of making this favorable tradeoff between the two error rates, while not redesigning the code and only trivially modifying the decoder.

This work was done by Samuel Dolinar, Kenneth Andrews, Fabrizio Pollara, and Dariush Divsalar of Caltech for NASA’s Jet Propulsion Laboratory.

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