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Electronic Colloquium on Computational Complexity

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Reports tagged with List Decoding Capacity:
TR05-133 | 17th November 2005
Venkatesan Guruswami, Atri Rudra

Explicit Capacity-Achieving List-Decodable Codes

Revisions: 1

For every $0 < R < 1$ and $\eps > 0$, we present an explicit
construction of error-correcting codes of rate $R$ that can be list
decoded in polynomial time up to a fraction $(1-R-\eps)$ of errors.
These codes achieve the ``capacity'' for decoding from {\em ... more >>>

TR07-109 | 7th October 2007
Venkatesan Guruswami, Atri Rudra

Better Binary List-Decodable Codes via Multilevel Concatenation

We give a polynomial time construction of binary codes with the best
currently known trade-off between rate and error-correction
radius. Specifically, we obtain linear codes over fixed alphabets
that can be list decoded in polynomial time up to the so called
Blokh-Zyablov bound. Our work ... more >>>

TR08-054 | 13th May 2008
Venkatesan Guruswami, Atri Rudra

Concatenated codes can achieve list-decoding capacity

We prove that binary linear concatenated codes with an outer algebraic code (specifically, a folded Reed-Solomon code) and independently and randomly chosen linear inner codes achieve the list-decoding capacity with high probability. In particular, for any $0 < \rho < 1/2$ and $\epsilon > 0$, there exist concatenated codes of ... more >>>

TR19-122 | 13th September 2019
Jonathan Mosheiff, Nicolas Resch, Noga Ron-Zewi, Shashwat Silas, Mary Wootters

LDPC Codes Achieve List-Decoding Capacity

Revisions: 2

We show that Gallager's ensemble of Low-Density Parity Check (LDPC) codes achieve list-decoding capacity. These are the first graph-based codes shown to have this property. Previously, the only codes known to achieve list-decoding capacity were completely random codes, random linear codes, and codes constructed by algebraic (rather than combinatorial) techniques. ... more >>>

TR20-179 | 2nd December 2020
Siddharth Bhandari, Prahladh Harsha, Mrinal Kumar, Madhu Sudan

Decoding Multivariate Multiplicity Codes on Product Sets

The multiplicity Schwartz-Zippel lemma bounds the total multiplicity of zeroes of a multivariate polynomial on a product set. This lemma motivates the multiplicity codes of Kopparty, Saraf and Yekhanin [J. ACM, 2014], who showed how to use this lemma to construct high-rate locally-decodable codes. However, the algorithmic results about these ... more >>>

TR21-036 | 14th March 2021
Siddharth Bhandari, Prahladh Harsha, Mrinal Kumar, Madhu Sudan

Ideal-theoretic Explanation of Capacity-achieving Decoding

In this work, we present an abstract framework for some algebraic error-correcting codes with the aim of capturing codes that are list-decodable to capacity, along with their decoding algorithm. In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and their ... more >>>

TR21-139 | 24th September 2021
Venkatesan Guruswami, Jonathan Mosheiff

Punctured Large Distance Codes, and Many Reed-Solomon Codes, Achieve List-Decoding Capacity

We prove the existence of Reed-Solomon codes of any desired rate $R \in (0,1)$ that are combinatorially list-decodable up to a radius approaching $1-R$, which is the information-theoretic limit. This is established by starting with the full-length $[q,k]_q$ Reed-Solomon code over a field $\mathbb{F}_q$ that is polynomially larger than the ... more >>>

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