Klim Efremenko

3-query locally decodable codes of subexponential length

Locally Decodable Codes (LDC) allow one to decode any particular symbol of the input message by making a constant number of queries to a codeword, even if a constant fraction of the codeword is damaged. In a recent work [Yek08] Yekhanin constructs a 3-query LDC with sub-exponential length of size exp(exp(O((log n)/(log log n)))). However, this construction requires a conjecture that there are infinitely many Mersenne primes. In this paper we give the first unconditional constant query LDC construction with subexponantial codeword length. In addition our construction reduces codeword length. We give construction of 3-query LDC with codeword length exp(exp(O(√{log n log log n ))). Our construction could also be extended to higher number of queries. We give a 2r-query LDC with length of exp(exp(O(☂[r] log n (log log n)r-1))).

List and Unique Coding for Interactive Communication in the Presence of Adversarial Noise

In this paper we extend the notion of list-decoding to the setting of interactive communication and study its limits. In particular, we show that any protocol can be encoded, with a constant rate, into a list-decodable protocol which is resilient to a noise rate of up to 1/2--e, and that this is tight. Using our list-decodable construction, we study a more nuanced model of noise where the adversary can corrupt up to a fraction α Alices communication and up to a fraction β of Bobs communication. We use list-decoding in order to fully characterize the region RU of pairs (α β) for which unique decoding with a constant rate is possible. The region RU turns out to be quite unusual in its shape. In particular, it is bounded by a piecewise-differentiable curve with infinitely many pieces. We show that outside this region, the rate must be exponential. This suggests that in some error regimes, list-decoding is necessary for optimal unique decoding. We also consider the setting where only one party of the communication must output the correct answer. We precisely characterize the region of all pairs (α β) for which one-sided unique decoding is possible in a way that Alice will output the correct answer.

3-Query Locally Decodable Codes of Subexponential Length

Locally decodable codes (LDCs) allow one to decode any particular symbol of the input message by making a constant number of queries to a codeword, even if a constant fraction of the codeword is damaged. In a recent work [J. ACM, 55 (2008), article 1], Yekhanin constructs a 3-query LDC with subexponential length. However, this construction requires a conjecture that there are infinitely many Mersenne primes. In this paper, we give the first unconditional constant query LDC construction with subexponential codeword length. In addition, our construction reduces codeword length.

Local List Decoding with a Constant Number of Queries

Recently Efremenko showed locally-decodable codes of sub-exponential length. That result showed that these codes can handle up to

A Black Box for Online Approximate Pattern Matching

\frac{1}{3}

Maximal Noise in Interactive Communication over Erasure Channels and Channels with Feedback

fraction of errors. In this paper we show that the same codes can be locally unique-decoded from error rate

Constant-rate coding for multiparty interactive communication is impossible

\half-\alpha

Maximal Noise in Interactive Communication Over Erasure Channels and Channels With Feedback

for any

From irreducible representations to locally decodable codes

\alpha>0

∊-MSR codes with small sub-packetization

and locally list-decoded from error rate