On the number of error correcting codes

Series
Combinatorics Seminar
Time
Friday, October 11, 2024 - 3:15pm for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nitya Mani – MIT – nmani@mit.eduhttps://www.mit.edu/~nmani/
Organizer
Xiaoyu He

We show that for a fixed q, the number of q-ary t-error correcting codes of length n is at most 2(1+o(1))Hq(n,t) for all t(1q1)n2nlogn, where Hq(n,t)=qn/Vq(n,t) is the Hamming bound and Vq(n,t) is the cardinality of the radius t Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner and makes progress towards a 2005 question of Sapozhenko.