The minimal distance of random linear codes

Stochastics Seminar
Thursday, November 7, 2019 - 3:05pm for 1 hour (actually 50 minutes)
Skiles 006
Han Huang – GeorgiaTech
Konstantin Tikhomirov

When Alice wants to send a k-bits message v to Bob over a noisy channel, she encodes it as a longer n-bits message Mv, where M is a n times k matrix over F_2. The minimal distance d_M of the linear code M is defined as the minimum Hamming distance between Mw and Mu over all distinct points w,u in F_2^k. In this way, if there are less than d_M/2 corrupted bits in the message, Bob can recover the original message via a nearest neighbor search algorithm.

The classical Gilbert-Varshamov Bound provides a lower bound for d_M if the columns of M are independent copies of X, where X is the random vector uniformly distributed on F_2^n. Under the same assumption on M, we show that the distribution of d_M is essentially the same as the minimum of Hamming weight (Hamming distance to origin) of 2^k-1 i.i.d copies of X.

The result is surprising since M is only generated by k independent copies of X. Furthermore, our results also work for arbitrary finite fields.

This is joint work with Jing Hao, Galyna Livshyts, Konstantin Tikhomirov.