Posted by: atri | March 18, 2009

## Lecture 23: Elias-Bassalygo bound

In today’s lecture, we proved the following upper bound on rate called the Elias-Bassalygo bound: $R\le 1-H_q(J_q(\delta))+o(1)$, where $J_q(\delta)=\left(1-\frac{1}{q}\right)\sqrt{1-\frac{q}{q-1}\cdot\delta}$. The proof followed almost immediately from the Johnson bound and a simple argument to bound the average list size over random received words (which we had used earlier to prove the converse of the list decoding capacity).

Later on we took stock of the various open questions, including the question of whether the GV bound is tight. We fleetingly saw a mention of the algebraic-geometric codes, which beat the GV bound for alphabet size $q\ge 49$.

The material covered in today’s lecture corrresponds to the scribed notes for Lecture 19 from Fall 07. (The notes are not polished yet: I plan to polish them by this weekend.)

Next lecture, we will see why the GV bound is not tight for binary codes with constant distance. In particular, we will study BCH codes, which pretty much match the Hamming bound for any distance (we have seen the Hamming code, which matches the Hamming bound for $d=3$).  Depending on the time, we will also start looking at an application of BCH codes in construction of $\ell$-wise independent sources.