General Linearized Polynomial Interpolation and Its Applications

Xie, Hongwei; Yan, Zhiyuan; Suter, Bruce W.

doi:10.1109/isnetcod.2011.5978942

Cited by 19 publications

(40 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If n t h ≈ m then we may write τ f ≈ s 1 − 1/h+h h−s+1 R . Problem 1 can be solved by the efficient interpolation algorithm in [15] requiring at most O(s 2 n r D(h − s + 1)) < O(s 2 n 2 r ) operations in F q m .…”

Section: Lemma 2 a Nonzero Polynomial Fulfilling The Interpolation Comentioning

confidence: 99%

See 1 more Smart Citation

List and probabilistic unique decoding of folded subspace codes

Bartz

Sidorenko

2015

2015 IEEE International Symposium on Information Theory (ISIT)

View full text Add to dashboard Cite

Abstract-A new class of folded subspace codes for noncoherent network coding is presented. The codes can correct insertions and deletions beyond the unique decoding radius for any code rate R ∈ [0, 1]. An efficient interpolation-based decoding algorithm for this code construction is given which allows to correct insertions and deletions up to the normalized radius s (1 − ((1/h + h)/(h − s + 1))R), where h is the folding parameter and s ≤ h is a decoding parameter. The algorithm serves as a list decoder or as a probabilistic unique decoder that outputs a unique solution with high probability. An upper bound on the average list size of (folded) subspace codes and on the decoding failure probability is derived. A major benefit of the decoding scheme is that it enables probabilistic unique decoding up to the list decoding radius.

show abstract

Section: Lemma 2 a Nonzero Polynomial Fulfilling The Interpolation Comentioning

confidence: 99%

“…q is always in the column space of B since we guarantee that the transmitted message polynomial f (x) is a solution to (11) if γ and δ satisfy (10). Due to the lower triangular structure of B the root-finding system (15) can be solved in at most O(k 2 ) operations in F q m .…”

Section: B Root-finding Stepmentioning

confidence: 99%

List and probabilistic unique decoding of folded subspace codes

Bartz

Sidorenko

2015

2015 IEEE International Symposium on Information Theory (ISIT)

View full text Add to dashboard Cite

show abstract

“…(5), a general interpolation algorithm by linearized polynomials in [12] can be used, which requires O(k 2 ) multiplications and O(k 2 ) additions over GF(q m ). Hence, data reconstruction has a quadratic complexity with respect to the code dimension.…”

Section: Complexity Analysismentioning

confidence: 99%

MDS codes with low repair complexity for distributed storage networks

Xie

Yan

2013

2013 22nd Wireless and Optical Communication Conference

Self Cite

View full text Add to dashboard Cite

In this paper, we propose two new constructions of maximum distance separable (MDS) codes with low repair complexity for distributed storage networks. For both constructions, the encoded symbols are obtained by first treating the message vector as a linearized polynomial and then evaluating it at carefully chosen points. Compared to traditional MDS codes, data repair for our codes does not have to decode the entire original message, but only forms linear combinations of available encoded symbols. This linear repair complexity is attractive for applications with a large amount of data repair.

show abstract

“…However, this method does not take advantage of the certain structure of this system of equations and therefore, it is not efficient. An efficient polynomial-time interpolation algorithm in the ring of linearized polynomials is presented in [14] which is basically analogous to Koetter interpolation algorithm in the ring of polynomials. The factorization step can be efficiently done in polynomial-time using Roth-Ruckenstein algorithm [12].…”

Section: ) Factorizationmentioning

confidence: 99%

Algebraic list-decoding of subspace codes with multiplicities

Mahdavifar

Vardy

2011

2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

View full text Add to dashboard Cite

Koetter and Kschischang introduced subspace codes in order to correct errors and erasures for randomized network coding, in the case where network topology is unknown (the noncoherent case). The codewords are vector subspaces of a fixed ambient space; thus codes for this model are collections of such subspaces. Koetter and Kschischang constructed a remarkable family of codes similar to Reed-Solomon codes in that codewords are obtained by evaluating certain (linearized) polynomials.In a previous work, we introduced a new family of subspace codes based upon Koetter-Kschichang construction wich are efficiently list-decodable. By that we could achieve a better decoding radius at low rates than Koetter-Kschischiang codes. In this paper, we consider the problem of list-decoding of subspace codes with multiplicities. We first establish the notion of multiplicity for linearized polynomials in this context. Then we take advantage of enforcing multiple roots for the interpolation polynomial in order to achieve a better decoding radius. We are also able to list-decode at higher rates. The end result is the following: for any integers L and r, our list-L decoder with multiplicity r guarantees successful recovery of the message subspace provided that the normalized dimension of the error is at mostwhere R * is the normalized packet rate. This improves the normalized decoding radius upon the previous results for a wide range of rates. The parameter r is independent of the code construction and can be chosen at the decoder in such a way that the decoding radius is maximized. As L tends to infinity, the decoding radius of our construction with appropriate choice of r approaches 1 R * − 1.

show abstract

General Linearized Polynomial Interpolation and Its Applications

Cited by 19 publications

References 20 publications

List and probabilistic unique decoding of folded subspace codes

List and probabilistic unique decoding of folded subspace codes

MDS codes with low repair complexity for distributed storage networks

Algebraic list-decoding of subspace codes with multiplicities

Contact Info

Product

Resources

About