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Computer Science

Fast Ecoding/Decoding Algorithms for Reed-Solomon Erasure Codes

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Abstract

In this paper, we present a new basis of polynomial over finite fields of characteristic two and then apply it to the encoding/decoding of Reed-Solomon erasure codes. The proposed polynomial basis allows that h-point polynomial evaluation can be computed in O(h log 2 (h)) finite field operations with small leading constant. As compared with the canonical polynomial basis, the proposed basis improves the arithmetic complexity of addition, multiplication, and the determination of polynomial degree from O(h log 2 (h) log 2 log 2 (h)) to O(h log 2 (h)). Based on this basis, we then develop the encoding and erasure decoding algorithms for the (n = 2 r , k) Reed-Solomon codes. Thanks to the efficiency of transform based on the polynomial basis, the encoding can be completed in O(n log 2 (k)) finite field operations, and the erasure decoding in O(n log 2 (n)) finite field operations. To the best of our knowledge, this is the first approach supporting Reed-Solomon erasure codes over characteristic-2 finite fields while achieving a complexity of O(n log 2 (n)), in both additive and multiplicative complexities. As the complexity leading factor is small, the algorithms are advantageous in practical applications.

Key takeaways
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  1. This work achieves O(n log2(n)) complexity for encoding/decoding Reed-Solomon codes over characteristic-2 fields.
  2. The new polynomial basis reduces arithmetic complexity from O(h log2(h) log2 log2(h)) to O(h log2(h)).
  3. Encoding requires O(n log2(k)) operations while erasure decoding takes O(n log2(n)) operations.
  4. The proposed algorithms significantly outperform existing approaches, showing a 17x speedup in practical applications.
  5. The findings suggest potential for parallel processing in Reed-Solomon algorithms, enhancing efficiency further.
Figures (3)
Fig. 1. Data flow diagram of proposed transform of length h = 8. In this section, we consider the formal derivative over the proposed basis. Section IV-A gives the closed form of the formal derivative. SectionIV-B presents a computation method that has lower multiplicative complexity than the original approach.
Fig. 1. Data flow diagram of proposed transform of length h = 8. In this section, we consider the formal derivative over the proposed basis. Section IV-A gives the closed form of the formal derivative. SectionIV-B presents a computation method that has lower multiplicative complexity than the original approach.
COMPLEXITIES OF OPERATIONS IN POLYNOMIAL BASIS OVER CHARACTERISTIC-2 FINITE FIELDS
COMPLEXITIES OF OPERATIONS IN POLYNOMIAL BASIS OVER CHARACTERISTIC-2 FINITE FIELDS
TABLE II presented here for the purpose of completeness. Consider the construction of II’. The formal derivative of HI(a) is given by
TABLE II presented here for the purpose of completeness. Consider the construction of II’. The formal derivative of HI(a) is given by

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FAQs

sparkles

AI

What computational complexity does the proposed encoding algorithm achieve for Reed-Solomon codes?add

The proposed encoding algorithm achieves a computational complexity of O(n lg(n)), enabling efficient processing of Reed-Solomon codes.

How does the new polynomial basis improve polynomial evaluations over characteristic-2 fields?add

The new polynomial basis allows polynomial evaluations in O(h lg(h)) time, which is more efficient than previous methods.

What role does the formal derivative play in the decoding algorithm for Reed-Solomon codes?add

The formal derivative is computed to aid in reconstructing the erased symbols, ensuring efficient decoding processes.

How do the proposed algorithms compare with existing Reed-Solomon decoding methods?add

The proposed algorithms are approximately 17 times faster than existing methods, improving overall decoding efficiency on characteristic-2 finite fields.

What supports the claim that the new method is the first to achieve O(n lg(n)) complexity?add

This assertion is based on empirical evidence showing no prior method has lower decoding complexity for Reed-Solomon codes over characteristic-2 fields.

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Sian-Jheng Lin

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In this paper, we present a new basis of polynomial over finite fields of characteristic two and then apply it to the encoding/decoding of Reed-Solomon erasure codes. The proposed polynomial basis allows that h-point polynomial evaluation can be computed in O(h log 2 (h)) finite field operations with small leading constant. As compared with the canonical polynomial basis, the proposed basis improves the arithmetic complexity of addition, multiplication, and the determination of polynomial degree from O(h log 2 (h) log 2 log 2 (h)) to O(h log 2 (h)). Based on this basis, we then develop the encoding and erasure decoding algorithms for the (n = 2 r , k) Reed-Solomon codes. Thanks to the efficiency of transform based on the polynomial basis, the encoding can be completed in O(n log 2 (k)) finite field operations, and the erasure decoding in O(n log 2 (n)) finite field operations. To the best of our knowledge, this is the first approach supporting Reed-Solomon erasure codes over characteristic-2 finite fields while achieving a complexity of O(n log 2 (n)), in both additive and multiplicative complexities. As the complexity leading factor is small, the algorithms are advantageous in practical applications.

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A new decoding algorithm for correcting both erasures and errors of reed-solomon codes
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In this paper, a high efficient decoding algorithm is developed here in order to correct both erasures and errors for Reed-Solomon (RS) codes based on the Euclidean algorithm together with the Berlekamp-Massey (BM) algorithm. The new decoding algorithm computes the errata locator polynomial and the errata evaluator polynomial simultaneously without performing polynomial divisions, and there is no need for the computation of the discrepancies and the field element inversions. Also, the separate computation of the Forney syndrome needed in the decoder is completely avoided. As a consequence, the complexity of this new decoding algorithm is dramatically reduced. Finally, the new algorithm has been verified through a software simulation using C ++ language. An illustrative example of (255,239) RS code using this program shows that the speed of the decoding process is approximately three times faster than that of the inverse-free Berlekamp-Massey algorithm.

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Sian-Jheng Lin

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