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Theory Seminar: Binary Codes with Resilience Beyond 1/4 via Interaction
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Klim Efremenko (Ben-Gurion University)
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Wednesday, 15.6.2022, 12:30
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Taub 201
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 a constant rate and requires Bob to only send one short message. 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. Joint work with Gillat Kol, Raghuvansh R. Saxena and Zhijun Zhang
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