### Wednesday, January 20, 2016 - 12:30pm to 1:00pm

### Location:

ASA Conference Room 6115 Gates & Hillman Centers### Speaker:

JOSHUA BRAKENSIEK, Undergraduate Student /JOSHUA%20BRAKENSIEKWe consider the problem of constructing binary codes to recover from k--bit deletions with efficient encoding/decoding, for a fixed k. The single deletion case is well understood, with the Varshamov-Tenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with \approx 2^n/n codewords of length n, i.e., at most log(n) bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than n^\Omega(1). For any fixed k, we construct a binary code with c_k log(n) redundancy that can be decoded from k deletions in O_k(n log^4(n)) time. The coefficient c_k can be taken to be O(k^2 log(k)), which is only quadratically worse than the optimal, non-constructive bound of O(k). We also indicate how to modify this code to allow for a combination of up to k insertions and deletions. We also note that among linear codes capable of correcting k deletions, the (k+1)-fold repetition code is essentially the best possible. Based on joint work with Venkatesan Guruswami and Samuel Zbarsky. — oshua Brakensiek is an undergraduate in the Department of Mathematical Sciences at Carnegie Mellon University. He has conducted research in topics such as coding theory, hardness of approximation, and computational astrophysics. Talks from SODA 2016