Posted by: atri | November 7, 2007

Lecture 31: Concatenated codes achieve the GV bound

In today’s lecture we started to prove Thommesen’s result which states that concatenating RS code with random independent inner linear binary codes, result in codes that with high probability lies on the GV bound.

Towards the end of the lecture, I got stuck with some of the geometric interpretations. In particular we were dealing with the functions $\alpha(z)=1-H(1-2^{z-1})$ and $f_x(\theta)=(1-\theta)^{-1}H^{-1}(1-\theta x)$. I claimed the following facts without proof:

1. The line segment between $(x,0)$ and $(\alpha(x),H^{-1}(1-\alpha(x))$ is tangent to the curve $H^{-1}(1-r)$ at $r=\alpha(x)$.
2. The intercept of the line segment between $(x,0)$ and $(\theta x,H^{-1}(1-\theta x))$ on the “y”-axis is exactly $f_x(\theta)$.

For proofs of these two facts, look at proof of Lemma 5.3 in this chapter of my thesis (the chapter also has a picture that I drew on the board today). The proof presented is for the general alphabet case.

Michel and Than asked why $f_x(\theta)\le \frac{1}{2}$ ? This inequalitydoes not hold when every $0\le \theta\le 1$. However, it does hold when $\theta\le \alpha(x)/x$, which is the range of $\theta$ that we care about. The proof that $f_x(\theta)\le \frac{1}{2}$ for $0\le\theta\le \alpha(x)/x$ follows from 1. and 2. above along with the fact that $H^{-1}(1-r)$ is a decreasing and strictly convex function of $r$.

Use the comments section, if you have any questions on this.

Next lecture, we will continue with Thommesen’s proof.