Folded algebraic-geometric codes from Galois extensions

Huang, Ming-Deh A.; Narayanan, Anand Kumar

doi:10.1090/conm/518/10209

Cited by 6 publications

(6 citation statements)

References 15 publications

(43 reference statements)

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“…It is mentioned that this algorithm is very fast experimentally and almost never explores too many candidate solutions. A similar approach was also considered in [17] for folded versions of algebraic-geometric codes. However, theoretically, it has not been possible to derive any polynomial guarantees on the size of the list returned by this approach or its running time (the obvious issue is that in each step there may be more than one candidate value of , leading to an exponential product bound on the runtime).…”

Section: Some Remarksmentioning

confidence: 99%

Linear-Algebraic List Decoding for Variants of Reed–Solomon Codes

Guruswami

Wang

2013

IEEE Trans. Inform. Theory

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Folded Reed-Solomon (RS) codes are an explicit family of codes that achieve the optimal tradeoff between rate and list error-correction capability: specifically, for any , Guruswami and Rudra presented an time algorithm to list decode appropriate folded RS codes of rate from a fraction of errors. The algorithm is based on multivariate polynomial interpolation and root-finding over extension fields. It was noted by Vadhan that interpolating a linear polynomial suffices for a statement of the above form. Here, we give a simple linear-algebra-based analysis of this variant that eliminates the need for the computationally expensive root-finding step over extension fields (and indeed any mention of extension fields). The entire list-decoding algorithm is linear-algebraic, solving one linear system for the interpolation step, and another linear system to find a small subspace of candidate solutions. Except for the step of pruning this subspace, the algorithm can be implemented to run in quadratic time. We also consider a closely related family of codes, called (order ) derivative codes and defined over fields of large characteristic, which consist of the evaluations of as well as its first formal derivatives at distinct field elements. We show how our linear-algebraic methods for folded RS codes can be used to show that derivative codes can also achieve the above optimal tradeoff. The theoretical drawback of our analysis for folded RS codes and derivative codes is that both the decoding complexity and proven worst-case list-size bound are . By combining the above idea with a pseudorandom subset of all polynomials as messages, we get a Monte Carlo construction achieving a list-size bound of which is quite close to the existential bound (however, the decoding complexity remains ). Our work highlights that constructing an explicit subspace-evasive subset that has small intersection with low-dimensional subspaces-an interesting problem in pseudorandomness in its own right-could lead to explicit codes with better list-decoding guarantees.

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Section: Some Remarksmentioning

confidence: 99%

Linear-Algebraic List Decoding for Variants of Reed–Solomon Codes

Guruswami

Wang

2013

IEEE Trans. Inform. Theory

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“…It is mentioned that this algorithm is very fast experimentally and almost never explores too many candidate solutions. A similar approach was also considered in [16] for folded versions of algebraic-geometric codes. However, theoretically it has not been possible to derive any polynomial guarantees on the size of the list returned by this approach or its running time (the obvious issue is that in each step there may be more than one candidate value of f i , leading to an exponential product bound on the runtime).…”

Section: Linear (Instead Of Affine) Space Of Solutionsmentioning

confidence: 99%

Linear-Algebraic List Decoding of Folded Reed-Solomon Codes

Guruswami

2011

2011 IEEE 26th Annual Conference on Computational Complexity

103

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Folded Reed-Solomon codes are an explicit family of codes that achieve the optimal tradeoff between rate and error-correction capability: specifically, for any ε > 0, the author and Rudra (2006,08) presented an n O(1/ε) time algorithm to list decode appropriate folded RS codes of rate R from a fraction 1 − R − ε of errors. The algorithm is based on multivariate polynomial interpolation and root-finding over extension fields. It was noted by Vadhan that interpolating a linear polynomial suffices if one settles for a smaller decoding radius (but still enough for a statement of the above form). Here we give a simple linear-algebra based analysis of this variant that eliminates the need for the computationally expensive root-finding step over extension fields (and indeed any mention of extension fields). The entire list decoding algorithm is linear-algebraic, solving one linear system for the interpolation step, and another linear system to find a small subspace of candidate solutions. Except for the step of pruning this subspace, the algorithm can be implemented to run in quadratic time.The theoretical drawback of folded RS codes are that both the decoding complexity and proven worst-case list-size bound are n Ω(1/ε) . By combining the above idea with a pseudorandom subset of all polynomials as messages, we get a Monte Carlo construction achieving a list size bound of O(1/ε 2 ) which is quite close to the existential O(1/ε) bound (however, the decoding complexity remains n Ω(1/ε) ).Our work highlights that constructing an explicit subspace-evasive subset that has small intersection with low-dimensional subspaces -an interesting problem in pseudorandomness in its own right -could lead to explicit codes with better list-decoding guarantees.

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“…Independent of our work, Huang and Narayanan [2008] have considered AG codes constructed from Galois extensions, and observed how automorphisms of large order can be used for folding such codes. To our knowledge, the only instantiation of this approach that improves on folded RS codes is the one based on cyclotomic function fields from our work.…”

Section: Venkatesan Guruswamimentioning

confidence: 99%

Untitled

2010

ANT

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This paper is dedicated to Siegfried Bosch, whose foundational work in rigid analysis was invaluable in our development of the theory of semistable coverings. We determine the stable models of the modular curves X 0 (p 3) for primes p ≥ 13. An essential ingredient is the close relationship between the deformation theories of elliptic curves and formal groups, which was established in the Woods Hole notes of 1964. This enables us to apply results of Hopkins and Gross in our analysis of the supersingular locus.

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Folded algebraic-geometric codes from Galois extensions

Cited by 6 publications

References 15 publications

Linear-Algebraic List Decoding for Variants of Reed–Solomon Codes

Linear-Algebraic List Decoding for Variants of Reed–Solomon Codes

Linear-Algebraic List Decoding of Folded Reed-Solomon Codes

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