The algorithm will not be explained here, but the following references were helpful * in creating this implementation:
* *Much credit is due to William Rucklidge since portions of this code are an indirect * port of his C++ Reed-Solomon implementation.
* * @author Sean Owen * @author William Rucklidge * @author sanfordsquires */ final class ReedSolomonDecoder { public function __construct(private $field) { } /** *Decodes given set of received codewords, which include both data and error-correction * codewords. Really, this means it uses Reed-Solomon to detect and correct errors, in-place, * in the input.
* * @param data $received and error-correction codewords * @param number $twoS of error-correction codewords available * * @throws ReedSolomonException if decoding fails for any reason */ public function decode(&$received, $twoS) { $poly = new GenericGFPoly($this->field, $received); $syndromeCoefficients = fill_array(0, $twoS, 0); $noError = true; for ($i = 0; $i < $twoS; $i++) { $eval = $poly->evaluateAt($this->field->exp($i + $this->field->getGeneratorBase())); $syndromeCoefficients[(is_countable($syndromeCoefficients) ? count($syndromeCoefficients) : 0) - 1 - $i] = $eval; if ($eval != 0) { $noError = false; } } if ($noError) { return; } $syndrome = new GenericGFPoly($this->field, $syndromeCoefficients); $sigmaOmega = $this->runEuclideanAlgorithm($this->field->buildMonomial($twoS, 1), $syndrome, $twoS); $sigma = $sigmaOmega[0]; $omega = $sigmaOmega[1]; $errorLocations = $this->findErrorLocations($sigma); $errorMagnitudes = $this->findErrorMagnitudes($omega, $errorLocations); $errorLocationsCount = is_countable($errorLocations) ? count($errorLocations) : 0; for ($i = 0; $i < $errorLocationsCount; $i++) { $position = (is_countable($received) ? count($received) : 0) - 1 - $this->field->log($errorLocations[$i]); if ($position < 0) { throw new ReedSolomonException("Bad error location"); } $received[$position] = GenericGF::addOrSubtract($received[$position], $errorMagnitudes[$i]); } } private function runEuclideanAlgorithm($a, $b, $R) { // Assume a's degree is >= b's if ($a->getDegree() < $b->getDegree()) { $temp = $a; $a = $b; $b = $temp; } $rLast = $a; $r = $b; $tLast = $this->field->getZero(); $t = $this->field->getOne(); // Run Euclidean algorithm until r's degree is less than R/2 while ($r->getDegree() >= $R / 2) { $rLastLast = $rLast; $tLastLast = $tLast; $rLast = $r; $tLast = $t; // Divide rLastLast by rLast, with quotient in q and remainder in r if ($rLast->isZero()) { // Oops, Euclidean algorithm already terminated? throw new ReedSolomonException("r_{i-1} was zero"); } $r = $rLastLast; $q = $this->field->getZero(); $denominatorLeadingTerm = $rLast->getCoefficient($rLast->getDegree()); $dltInverse = $this->field->inverse($denominatorLeadingTerm); while ($r->getDegree() >= $rLast->getDegree() && !$r->isZero()) { $degreeDiff = $r->getDegree() - $rLast->getDegree(); $scale = $this->field->multiply($r->getCoefficient($r->getDegree()), $dltInverse); $q = $q->addOrSubtract($this->field->buildMonomial($degreeDiff, $scale)); $r = $r->addOrSubtract($rLast->multiplyByMonomial($degreeDiff, $scale)); } $t = $q->multiply($tLast)->addOrSubtract($tLastLast); if ($r->getDegree() >= $rLast->getDegree()) { throw new ReedSolomonException("Division algorithm failed to reduce polynomial?"); } } $sigmaTildeAtZero = $t->getCoefficient(0); if ($sigmaTildeAtZero == 0) { throw new ReedSolomonException("sigmaTilde(0) was zero"); } $inverse = $this->field->inverse($sigmaTildeAtZero); $sigma = $t->multiply($inverse); $omega = $r->multiply($inverse); return [$sigma, $omega]; } private function findErrorLocations($errorLocator) { // This is a direct application of Chien's search $numErrors = $errorLocator->getDegree(); if ($numErrors == 1) { // shortcut return [$errorLocator->getCoefficient(1)]; } $result = fill_array(0, $numErrors, 0); $e = 0; for ($i = 1; $i < $this->field->getSize() && $e < $numErrors; $i++) { if ($errorLocator->evaluateAt($i) == 0) { $result[$e] = $this->field->inverse($i); $e++; } } if ($e != $numErrors) { throw new ReedSolomonException("Error locator degree does not match number of roots"); } return $result; } private function findErrorMagnitudes($errorEvaluator, $errorLocations) { // This is directly applying Forney's Formula $s = is_countable($errorLocations) ? count($errorLocations) : 0; $result = fill_array(0, $s, 0); for ($i = 0; $i < $s; $i++) { $xiInverse = $this->field->inverse($errorLocations[$i]); $denominator = 1; for ($j = 0; $j < $s; $j++) { if ($i != $j) { //denominator = field.multiply(denominator, // GenericGF.addOrSubtract(1, field.multiply(errorLocations[j], xiInverse))); // Above should work but fails on some Apple and Linux JDKs due to a Hotspot bug. // Below is a funny-looking workaround from Steven Parkes $term = $this->field->multiply($errorLocations[$j], $xiInverse); $termPlus1 = ($term & 0x1) == 0 ? $term | 1 : $term & ~1; $denominator = $this->field->multiply($denominator, $termPlus1); } } $result[$i] = $this->field->multiply( $errorEvaluator->evaluateAt($xiInverse), $this->field->inverse($denominator) ); if ($this->field->getGeneratorBase() != 0) { $result[$i] = $this->field->multiply($result[$i], $xiInverse); } } return $result; } }