TY - GEN

T1 - Binary Codes with Resilience Beyond 1/4 via Interaction

AU - Efremenko, Klim

AU - Kol, Gillat

AU - Saxena, Raghuvansh R.

AU - Zhang, Zhijun

N1 - Funding Information:
Klim Efremenko is supported by the Israel Science Foundation (ISF) through grant No. 1456/18 and European Research Council Grant number: 949707. Gillat Kol is supported by a National Science Foundation CAREER award CCF-1750443 and by a BSF grant No. 2018325.
Publisher Copyright:
© 2022 IEEE.

PY - 2022

Y1 - 2022

N2 - In the reliable transmission problem, a sender, Alice, wishes to transmit a bit-string x to a remote receiver, Bob, over a binary channel with adversarial noise. The solution to this problem is to encode x using an error correcting code. As it is long known that the distance of binary codes is at most 1/2, reliable transmission is possible only if the channel corrupts (flips) at most a 1/4-fraction of the communicated bits.We revisit the reliable transmission problem in the two-way setting, where both Alice and Bob can send bits to each other. Our main result is the construction of two-way error correcting codes that are resilient to a constant fraction of corruptions strictly larger than 1/4. Moreover, our code has constant rate and requires Bob to only send one short message. We mention that our result resolves an open problem by Haeupler, Kamath, and Velingker [APPROX-RANDOM, 2015] and by Gupta, Kalai, and Zhang [STOC, 2022].Curiously, our new two-way code requires a fresh perspective on classical error correcting codes: While classical codes have only one distance guarantee for all pairs of codewords (i.e., the minimum distance), we construct codes where the distance between a pair of codewords depends on the 'compatibility' of the messages they encode. We also prove that such codes are necessary for our result.

AB - In the reliable transmission problem, a sender, Alice, wishes to transmit a bit-string x to a remote receiver, Bob, over a binary channel with adversarial noise. The solution to this problem is to encode x using an error correcting code. As it is long known that the distance of binary codes is at most 1/2, reliable transmission is possible only if the channel corrupts (flips) at most a 1/4-fraction of the communicated bits.We revisit the reliable transmission problem in the two-way setting, where both Alice and Bob can send bits to each other. Our main result is the construction of two-way error correcting codes that are resilient to a constant fraction of corruptions strictly larger than 1/4. Moreover, our code has constant rate and requires Bob to only send one short message. We mention that our result resolves an open problem by Haeupler, Kamath, and Velingker [APPROX-RANDOM, 2015] and by Gupta, Kalai, and Zhang [STOC, 2022].Curiously, our new two-way code requires a fresh perspective on classical error correcting codes: While classical codes have only one distance guarantee for all pairs of codewords (i.e., the minimum distance), we construct codes where the distance between a pair of codewords depends on the 'compatibility' of the messages they encode. We also prove that such codes are necessary for our result.

KW - error correcting code

KW - interactive communication

KW - noise resilience

UR - http://www.scopus.com/inward/record.url?scp=85144771200&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85144771200&partnerID=8YFLogxK

U2 - 10.1109/FOCS54457.2022.00008

DO - 10.1109/FOCS54457.2022.00008

M3 - Conference contribution

AN - SCOPUS:85144771200

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 1

EP - 12

BT - Proceedings - 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science, FOCS 2022

PB - IEEE Computer Society

T2 - 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022

Y2 - 31 October 2022 through 3 November 2022

ER -