Posted by: atri | February 18, 2009

## Lecture 15: Plotkin Bound

Today we proved the Plotkin bound, modulo a geometric lemma that we will prove next lecture. Using the Plotkin bound we also obtained a new rate vs. distance tradeoff: $R\le 1-\left(\frac{q}{q-1}\right)\delta+o(1)$. This improves upon the Hamming bound for large enough $\delta$. Later on in the course we will also see the Elias-Bassalygo, which will a better upper bound than both the Hamming and Plotkin bounds. There are two reasons for considering the Elias-Bassalygo bound later in the course: the first is that I need to get more material done so that you guys have a decent coverage in your homework and second, one of the building block of the Elias-Bassalygo bound called the Johnson bound has a very natural implication for “list decoding,” which we are going to consider right after talking about Shannon’s theorem.

The material covered in today’s lecture can be found in the notes from the following fall 07 lectures: Lecture 16 and Lecture 17 (the latter still needs to be polished: I plan to do that  by Friday).