# Introduction to coding theory (CMU, Spr 2010)

## January 30, 2010

### Lecture 6 summary

Filed under: Lecture summary — Venkat Guruswami @ 3:29 pm

We finished the proof of the Plotkin upper bound ${1 - q\delta/(q-1)}$ on the rate of ${q}$-ary codes of relative distance ${\delta}$. We discussed the Johnson bound, and its implications for list decoding, a topic we will study in some detail later in the course. We mentioned the relation between the Johnson bound and bounds on ${A_q(n,d)}$ (you are asked to prove this for the binary case on your problem set). Using this and the proof of the Johnson bound for the binary case, we concluded the Elias-Bassalygo upper bound, our last of the “elementary” upper bounds.

In the next lecture, we will start our discussion of the linear programming bound, which remains the best known bound to date. We proved the following simple fact that plays a crucial role in the LP bound: For a linear code $C$, the quantity ${\sum_{c \in C} (-1)^{x \cdot c}}$ equals $|C|$ if $x \in C^\perp$ and equals $0$ otherwise.