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.

Existence theorem

Theorem: Let $\varepsilon$ > 0. For a large enough $k$, there exists an ensemble of inner codes $C_{in}^1,C_{in}^2,..,C_{in}^N$ of rate $\frac{1}{2}$, where $N = q^k - 1$, such that for at least $\left( {1 - \varepsilon } \right)N$ values of i, $C_{in}^i$ has relative distance $\ge H_q^{ - 1} (\frac{1}{2} - \varepsilon )$.

Here relative distance is the ratio of minimum distance to block length. And $H_q$ is the q-ary entropy function defined as follows: $H_q(x) = xlog_q(q-1)-xlog_qx-(1-x)log_q(1-x)$.

In fact, to show the existence of this set of linear codes, we will specify this ensemble explicitly as follows: for $\alpha \in \mathbb{F}_{q^k } - \{ 0\}$, the inner code $C_{in}^\alpha :\mathbb{F}_q^k \to \mathbb{F}_q^{2k}$, is defined as $C_{in}^\alpha (x) = (x,\alpha x)$. Here we can notice that $x \in \mathbb{F}_q^k$ and $\alpha \in \mathbb{F}_{q^k}$. We can do the multiplication $\alpha x$ since $\mathbb{F}_q^k$ is isomorphic to $\mathbb{F}_{q^k}$.

This ensemble is due to Wozencraft and is called the Wozencraft ensemble.

For any x and y in $\mathbb{F}_q^k$, we have the following facts:

1. $C_{in}^\alpha (x) + C_{in}^\alpha (y) = (x,\alpha x)+(y,\alpha y) = (x + y,\alpha (x + y)) = C_{in}^\alpha (x + y)$
2. For any $a \in F_q$, $aC_{in}^\alpha (x) = a(x,\alpha x) = \left( {ax,\alpha \left( {ax} \right)} \right) = C_{in}^\alpha (ax)$

So $C_{in}^\alpha$ is a linear code for every $\alpha \in \mathbb{F}_{q^k } - \{ 0\}$.

Now we know that Wozencraft ensemble contains linear codes with rate $\frac{1}{2}$. In the following proof, we will show that there are at least $\left( {1 - \varepsilon } \right)N$ those linear codes having the relative distance $\ge H_q^{ - 1} (\frac{1}{2} - \varepsilon )$, i.e. they meet the Gilbert-Varshamov bound.

Proof

To prove that there are at least $(1-\varepsilon)N$ number of linear codes in the Wozencraft ensemble having relative distance $\ge H_q^{-1}(\frac{1}{2}-\varepsilon)$, we will prove that there are at most $\varepsilon N$ number of linear codes having relative distance < $H_q^{-1}(\frac{1}{2}-\varepsilon)$ (i.e., having the distance < $H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k$).

Notice that in a linear code, the distance is equal to the minimum weight of all codewords of that code. This fact is the property of linear code. So if one non-zero codeword has the weight less than $H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k$, then that code has the distance less than $H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k$.

So $P$ = the number of linear codes having the distance less than $H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k$ = the number of linear codes having some codeword that has the weight less than $H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k$.

Now we have the following claim:

Claim: Two linear codes $C_{in}^{\alpha_1}$ and $C_{in}^{\alpha_2}$ (here $\alpha_1 \ne \alpha_2 \in \mathbb{F}_{q^k}-\{ 0 \}$) do not share any non-zero codeword.

Proof of Claim:

We prove the above claim by contradiction. Suppose there exist $\alpha_1 \ne \alpha_2 \in \mathbb{F}_{q^k}-\{ 0 \}$ such that two linear codes $C_{in}^{\alpha_1}$ and $C_{in}^{\alpha_2}$ contain the same non-zero codeword y.

Now since $y \in C_{in}^{\alpha_1}-\{ 0 \}$, $y = (y_1,\alpha_1 y_1)$ for some $y_1 \in \mathbb{F}_q^k$. As $y$ is non-zero, $y_1 \ne 0$.

Similarly, $y = (y_2,\alpha_2 y_2)$ for some $y_2 \in \mathbb{F}_q^k-\{ 0 \}$.

So $(y_1,\alpha_1 y_1) = (y_2,\alpha_2 y_2)$, then $y_1 = y_2 \ne 0$ and $\alpha_1 y_1 = \alpha_2 y_2$.

This implies $\alpha_1 = \alpha_2$, which is a contradiction, which completes the proof of the claim.

Now we come back to the proof of the theorem.

With any linear code having distance < $H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k$, it has some codeword that has the weight less than $H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k$.

Also due to the Claim, notice that no two linear code share the same non-zero codewords. This implies that if we have $P$ linear codes having distance < $H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k$, then we have at least $P$ different $y$ such that $wt(y)$ < $H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k$ (one such codeword $y$ for each linear code). Here $wt(y)$ denotes the weight of codeword $y$, which is the number of non-zero positions of $y$.

So $P$ (the number of linear codes having distance < $H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k$) is less than or equal the number of non-zero $y \in F_q^{2k}$ such that wt(y) < $H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k$.

Denote $S = \{ y|wt(y)$ < $H_q^{ - 1} (\frac{1}{2} - \varepsilon ) \cdot 2k\}$

So $P \le |S| \le Vol_q(H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k,2k)$

Here $Vol_q(r,n)$ is the Volume of Hamming ball of radius r in $[q]^n$.

Recall the upper bound of the Volume of Hamming ball $Vol_q(pn,n) \le q^{H_q(p )n}$ (check Bounds on the Volume of a Hamming ball for proof's detail), we have:

$P \le q^{H_q(H_q^{-1}(\frac{1}{2}-\varepsilon)) \cdot 2k} = q^{(\frac{1}{2}-\varepsilon) \cdot 2k} = \frac{q^k}{q^{2\varepsilon k}}$

When $k$ is large enough, we have $\frac{q^k}{q^{2\varepsilon k}}$ < $\varepsilon(q^k-1) = \varepsilon N$

So $P$ < $\varepsilon N$.

That means the number of linear codes having the relative distance < $H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k$ is less than $\varepsilon N$. So the number of linear codes having the relative distance at least $H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k$ is greater than $N - \varepsilon N = (1-\varepsilon)N$, which completes the proof.

See also

• Justesen code
• Linear code
• Hamming bound

References

• Massey, James L. (1963), Threshold decoding, Tech. Report 410, Cambridge, Mass.: Massachusetts Institute of Technology, Research Laboratory of Electronics, MR 0154763, hdl:1721.1/4415.
• Justesen, Jørn (1972), "A class of constructive asymptotically good algebraic codes", Institute of Electrical and Electronics Engineers. Transactions on Information Theory, IT-18: 652–656, doi:10.1109/TIT.1972.1054893, MR 0384313.

External links

• Lecture 28: Justesen Code. Coding theory's course. Prof. Atri Rudra.
• Lecture 9: Bounds on the Volume of a Hamming Ball. Coding theory's course. Prof. Atri Rudra.
• J. Justesen (1972). "A class of constructive asymptotically good algebraic codes". IEEE Trans. Info. Theory 18: 652–656. doi:10.1109/TIT.1972.1054893.
• Coding Theory's Notes: Gilbert-Varshamov Bound. Venkatesan Guruswami