Improved Decoding of Folded Reed-Solomon and Multiplicity Codes

Kopparty, Swastik; Ron-Zewi, Noga; Saraf, Shubhangi; Wootters, Mary

doi:10.1109/focs.2018.00029

Cited by 24 publications

(9 citation statements)

References 32 publications

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“…For |Σ| = 2, the list decoding capacity is ρ = 1 − H 2 (α) − γ, while for |Σ| = ω (1), the list decoding capacity is ρ = 1 − α − γ. Over large alphabets, this tradeoff can be achieved by explicit codes with efficient list decoding algorithms [21] (see also [27] for the state of the art). Over binary alphabet, we do not know how to explicitly construct codes achieving list decoding capacity.…”

Section: List Decodingmentioning

confidence: 99%

On List Recovery of High-Rate Tensor Codes

Kopparty

Resch

Ron-Zewi

et al. 2021

IEEE Trans. Inform. Theory

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We continue the study of list recovery properties of high-rate tensor codes, initiated by Hemenway, Ron-Zewi, and Wootters (FOCS'17). In that work it was shown that the tensor product of an efficient (poly-time) high-rate globally list recoverable code is approximately locally list recoverable, as well as globally list recoverable in probabilistic near-linear time. This was used in turn to give the first capacity-achieving list decodable codes with (1) local list decoding algorithms, and with (2) probabilistic near-linear time global list decoding algorithms. This also yielded constant-rate codes approaching the Gilbert-Varshamov bound with probabilistic near-linear time global unique decoding algorithms. In the current work we obtain the following results: 1. The tensor product of an efficient (poly-time) high-rate globally list recoverable code is globally list recoverable in deterministic near-linear time. This yields in turn the first capacity-achieving list decodable codes with deterministic near-linear time global list decoding algorithms. It also gives constant-rate codes approaching the Gilbert-Varshamov bound with deterministic near-linear time global unique decoding algorithms. 2. If the base code is additionally locally correctable, then the tensor product is (genuinely) locally list recoverable. This yields in turn (non-explicit) constant-rate codes approaching the Gilbert-Varshamov bound that are locally correctable with query complexity and running time N o(1). This improves over prior work by Gopi et. al. (SODA'17; IEEE Transactions on Information Theory'18) that only gave query complexity N ε with rate that is exponentially small in 1/ε. 3. A nearly-tight combinatorial lower bound on output list size for list recovering high-rate tensor codes. This bound implies in turn a nearly-tight lower bound of N Ω(1/ log log N) on the product of query complexity and output list size for locally list recovering high-rate tensor codes. 2012 ACM Subject Classification Mathematics of computing → Coding theory; Theory of computation → Pseudorandomness and derandomization Keywords and phrases Coding theory, Tensor codes, List-decoding and recovery, Local codes

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Section: List Decodingmentioning

confidence: 99%

On List Recovery of High-Rate Tensor Codes

Kopparty

Resch

Ron-Zewi

et al. 2021

IEEE Trans. Inform. Theory

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“…Sian et al [15] proposed using RS codes as an iterative structure and presented an algebraic notation for the input seeds. According to Swastik et al [16], the RS codes can be defined using two fields: GF(q) and GF(q m ); however, in this paper, GF(q m ) was chosen to give a better transmission. For example, the RS codes can transmit 128 random bits for a seven-bit input message sequence in the finite field 'F'.…”

Section: Related Work Of the Lfsrmentioning

confidence: 99%

“…The maximum length pseudo-random patterns with the minimum distances are examined using the concepts of multiplicity in the codes. The multiplicity structure in the codes should be identified using the multivariant low-degree polynomial and the restriction of the univariant low-degree polynomial [16]. Therefore, the polynomial for the LFSR is fixed as low-degree multivariant bits.…”

Section: Minimum Distances Achieved For the Maximum Length Pseudo-random Test Patternsmentioning

confidence: 99%

Design of a 1.9 GHz low-power LFSR Circuit using the Reed-Solomon Algorithm for Pseudo-Random Test Pattern Generation

Shivakumar

Senthilpari

Yusoff

2021

IJIE

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A linear feedback shift register (LFSR) has been frequently used in the Built-in Self-Test (BIST) designs for the pseudo-random test pattern generation. The higher volume of the test patterns and the lower test power consumption are the key features in the large complex designs. The motivation of this study is to generate efficient pseudo-random test patterns by the proposed LFSR and to be applied in the BIST designs. For the BIST designs, the proposed LFSR satisfied with the main strategies such as re-seeding and lesser test power consumption. However, the reseeding approach was utilized by the maximum-length pseudo-random test patterns. The objective of this paper is to propose a new LFSR circuit based on the proposed Reed-Solomon (RS) algorithm. The RS algorithm is created by considering the factors of the maximum length test patterns with a minimum distance over the time t. Also, it has been achieved an effective generation of test patterns over a stage of complexity order O (m log2 m), where m denotes the total number of message bits. We analysed our RS LFSR mathematically using the feedback polynomial function to decrease the area overhead occupied in the designs. The simulation works of the proposed RS LFSR bit-wise stages are simulated using the TSMC 130 nm on the Mentor Graphics IC design platform. Experimental results showed that the proposed LFSR achieved the effective pseudo-random test patterns with a lower test power consumption of 25.13 µW and 49.9 µs. In addition, proposed LFSR along with existing authors’ LFSR are applied in the BIST design to examine their power consumption. Ultimately, overall simulations operated with the highest operating frequency environment as 1.9 GHz.

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“…Furthermore, if we use a folded Reed-Solomon code, then we still get a Shamir-like threshold secret sharing scheme, but in addition we can employ list-decoding when we are in a regime were decoding is not unique anymore. In summary, we have the following (see Appendix A.1 and [12] for the details).…”

Section: Shamir Secret Sharing With List Decodingmentioning

confidence: 99%

“…There are many works aimed at reducing the list size of the folded Reed-Solomon codes. Recently, Kopparty et al [12] proved that the list size of the folded Reed-Solomon codes is at most a constant in γ.…”

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confidence: 99%

Towards Optimal Robust Secret Sharing with Security Against a Rushing Adversary

Fehr

Yuan

2019

Lecture Notes in Computer Science

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Robust secret sharing enables the reconstruction of a secretshared message in the presence of up to t (out of n) incorrect shares. The most challenging case is when n = 2t + 1, which is the largest t for which the task is still possible, up to a small error probability 2 −κ and with some overhead in the share size. Recently, Bishop, Pastro, Rajaraman and Wichs [3] proposed a scheme with an (almost) optimal overhead of O(κ). This seems to answer the open question posed by Cevallos et al. [6] who proposed a scheme with overhead of O(n + κ) and asked whether the linear dependency on n was necessary or not. However, a subtle issue with Bishop et al.'s solution is that it (implicitly) assumes a non-rushing adversary, and thus it satisfies a weaker notion of security compared to the scheme by Cevallos et al. [6], or to the classical scheme by Rabin and BenOr [13]. In this work, we almost close this gap. We propose a new robust secret sharing scheme that offers full security against a rushing adversary, and that has an overhead of O(κn ε), where ε > 0 is arbitrary but fixed. This n ε-factor is obviously worse than the polylog(n)-factor hidden in the O notation of the scheme of Bishop et al. [3], but it greatly improves on the linear dependency on n of the best known scheme that features security against a rushing adversary (when κ is substantially smaller than n). A small variation of our scheme has the same O(κ) overhead as the scheme of Bishop et al. and achieves security against a rushing adversary, but suffers from a (slightly) superpolynomial reconstruction complexity.

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Improved Decoding of Folded Reed-Solomon and Multiplicity Codes

Cited by 24 publications

References 32 publications

On List Recovery of High-Rate Tensor Codes

On List Recovery of High-Rate Tensor Codes

Design of a 1.9 GHz low-power LFSR Circuit using the Reed-Solomon Algorithm for Pseudo-Random Test Pattern Generation

Towards Optimal Robust Secret Sharing with Security Against a Rushing Adversary

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