Wozencraft ensemble

In coding theory, the Wozencraft ensemble is a set of linear codes in which most of codes satisfy the Gilbert-Varshamov bound. It is named after John Wozencraft, who proved its existence. The ensemble is described by Massey (1963), who attributes it to Wozencraft. Justesen (1972) used the Wozencraft ensemble as the inner codes in his construction of strongly explicit asymptotically good code.

Here relative distance is the ratio of minimum distance to block length. And ${\displaystyle H_{q}}$ is the q-ary entropy function defined as follows:

In fact, to show the existence of this set of linear codes, we will specify this ensemble explicitly as follows: for ${\displaystyle \alpha \in \mathbb {F} _{q^{k}}-\{0\}}$, define the inner code

Here we can notice that ${\displaystyle x\in \mathbb {F} _{q}^{k}}$ and ${\displaystyle \alpha \in \mathbb {F} _{q^{k}}}$. We can do the multiplication ${\displaystyle \alpha x}$ since ${\displaystyle \mathbb {F} _{q}^{k}}$ is isomorphic to ${\displaystyle \mathbb {F} _{q^{k}}}$.

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