A New Remainder Based Decoding Algorithm for Reed-Solomon Codes
Bibliographic record
Abstract
Conventional decoding techniques for decoding cyclic codes require the computation of power sum syndromes which can often account for a significant portion of the decoder computations. Since the syndromes can be computed from the remainder polynomial, the polynomial obtained by dividing the received polynomial by the code generator polynomial, it follows that this polynomial contains all the information required to decode. Thus one might hope for a decoding technique that uses the remainder polynomial directly. Berlekamp and Welch have given such an algorithm which requires the sequential testing of the parity check locations and updating of four polynomials. Whiting in his doctoral thesis has given a modification of this procedure that makes more efficient the evaluation and updating of these polynomials. The present work derives a new algorithm using only the remainder polynomial. A new key equation is derived which may be solved by the usual Euclidean algorithm. The advantages of this approach are discussed and compared to the original algorithm and a performance of the algorithm in terms of computational and circuit complexity is considered.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".