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The most effective decoding algorithms for cyclic codes are those, which

exploit the code structure and symmetry to the fullest possible extent, areextendable to soft-decision decoding and can offer trade-off between

decoding performance and complexity. Minimum weight decoding seems to

fulfill these requirements. It is based on the error trapping technique,

which is the simplest of the syndrome decoding methods, combined with

trial-and-error (systematic search) and step-by-step decoding methods,

resulting a very effective reduced-search bounded minimum-distance decodi~g

algorithm. It makes full use of the nature of a cyclic code and since it

is essentially a form of minimum-distance decoding, its adaption for

soft-decision decoding is quite simple, only requiring changing the metrics

to soft (Euclidean) distance from the hard (Hamming) distance. Itscomplexity is proportional to the square of the block length or less making

it ve.ry attractive for decoding short and medium length codes. Minimum

weight decoding, a burst-and-random-error-correcting algorithm, is also aninterpretative and unifying concept which explains the fundamental natureof syndrome decoding methods, and relates apparently distinct algorithms

within a coherent framework. It manifests a practical example of an

implementation of step-by-step decoding. In the trial-and-error stage of

the decoding algorithm there exists an upper bound on the maximum number of

trial-and-error positions necessary to be tested (i.e., not all

trial-and-error positions. need to be tried) which reduces the deocding

delay (or latency) and overall complexity quite considerably.

A study of the extension of hard- and soft-decision minimum weight decoding

to multi-level cyclic Reed-Solomon codes reveals that if thetrial-and-error stage of the basic decoding algorithm is modified, then for

example, with the RS (15,9) code, up to IdB in the hard-decision case and

2dB in the soft-decision case is gained, when bit errors, rather than

symbol errors, are corrected. Here the decoding algorithm can offer

trade-off between complexity and performance. For example, in the

soft-decision case the trial-and-error stage can be much simplified, and

therefore complexity reduced, at the price of IdB loss in the coding gain.The minimum weight decoding algorithm also can be extended to multi-level

- signalling schemes, such as ASK, m-ary PSK and QAM. On the two-dimensional

16-QA.\f A\\'GN channel, for example, simulation results show that the

soft-deci&ion minimum weight decoding of RS (15,9) code offers 1.5dB extra

coding gain over the hard-decision case and an overall coding gain of

2.5dB. This combined coding and modulat.ion scheme has very 10~' complexity.

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The MWD algorithm in the two-dimensional QAM channel, for example,

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Let the transmitter codeword be c'-and thc received signal be r at the decoder;

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G(x), multiplied by the appropriate field clement; using thc corresponding decision symbol in

the soft-decision case) thus forming a syndrome (the first finite field subtracter is also the first

candidate codcword); I

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(*) Calculate the Euclidcan distances between the symbols of r and the symbols of thc

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mctrics in each dimension; -!

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Calculate the "wcight", we . by adding up the Euclidean distances found in (*);

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respcctto the previous syndromc;V

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, : co.dcword, thus forming a new syndrome;

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If after a total of k + II shifts of G(.t), \Ve has never fallen to te or less, convert

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(*) Calculate the Euclidcan distanccs bclwccn lhc symbols of r and the symbols of the

candidate codcword;

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If we ~te (threshold detcrmincd cxpcrimcntally for HD and SD)

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stop;

else Shift G(x) cyclically (a total ofk + n timcs), and find thc ncxt

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eodcword, thus forming a ncw syndrome;

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