Elias Bassalygo bound

The Elias Bassalygo bound is a mathematical limit used in coding theory for error correction during data transmission or communications.

Definition
Let $$C$$ be a $$q$$-ary code of length $$n$$, i.e. a subset of $$[q]^n$$. Let $$R$$ be the rate of $$C$$, $$\delta$$ the relative distance and


 * $$B_q(y, \rho n) = \left \{ x \in [q]^n \ : \ \Delta(x, y) \leqslant \rho n \right \}$$

be the Hamming ball of radius $$ \rho n $$ centered at $$y$$. Let $$\text{Vol}_q(y, \rho n) = |B_q(y, \rho n)| $$ be the volume of the Hamming ball of radius $$ \rho n $$. It is obvious that the volume of a Hamming Ball is translation-invariant, i.e. indifferent to $$y.$$ In particular, $$ |B_q(y, \rho n)| =|B_q(0, \rho n)|.$$

With large enough $$n$$, the rate $$R$$ and the relative distance $$\delta$$ satisfy the Elias-Bassalygo bound:


 * $$R \leqslant 1 - H_q ( J_q(\delta))+o(1),$$

where


 * $$ H_q(x)\equiv_\text{def} -x\log_q \left ({x \over {q-1}} \right)-(1-x)\log_q{(1-x)} $$

is the q-ary entropy function and


 * $$J_q(\delta) \equiv_ \text{def} \left(1-{1\over q}\right)\left(1-\sqrt{1-{q \delta \over{q-1}}}\right) $$

is a function related with Johnson bound.

Proof
To prove the Elias–Bassalygo bound, start with the following Lemma:


 * Lemma. For $$C\subseteq [q]^n $$ and $$ 0\leqslant  e\leqslant  n$$, there exists a Hamming ball of radius $$e$$ with at least
 * $$\frac{|C|\text{Vol}_q(0,e)}{q^n}$$
 * codewords in it.


 * Proof of Lemma. Randomly pick a received word $$y \in [q]^n$$ and let $$B_q(y, 0)$$ be the Hamming ball centered at $$y$$ with radius $$e$$. Since $$y$$ is (uniform) randomly selected the expected size of overlapped region $$|B_q(y,e) \cap C|$$ is
 * $$\frac{|C|\text{Vol}_q(y,e)}{q^n}$$
 * Since this is the expected value of the size, there must exist at least one $$y$$ such that
 * $$|B_q(y,e) \cap C| \geqslant {{|C|\text{Vol}_q(y,e)} \over {q^n}} = {{|C|\text{Vol}_q(0,e)} \over {q^n}},$$
 * otherwise the expectation must be smaller than this value.

Now we prove the Elias–Bassalygo bound. Define $$e = n J_q(\delta)-1.$$ By Lemma, there exists a Hamming ball with $$B$$ codewords such that:


 * $$B\geqslant { {|C|\text{Vol}(0,e)} \over {q^n}} $$

By the Johnson bound, we have $$B\leqslant qdn$$. Thus,


 * $$| C | \leqslant qnd \cdot {{q^n} \over {\text{Vol}_q(0,e)}} \leqslant q^{n(1-H_q(J_q(\delta))+o(1))}$$

The second inequality follows from lower bound on the volume of a Hamming ball:


 * $$ \text{Vol}_q \left (0, \left \lfloor \frac{d-1}{2} \right \rfloor \right ) \geqslant q^{H_q \left (\frac{\delta}{2} \right )n-o(n)}.$$

Putting in $$d=2e+1$$ and $$ \delta = \tfrac{d}{n}$$ gives the second inequality.

Therefore we have


 * $$R={\log_q{|C|} \over n} \leqslant 1-H_q(J_q(\delta))+o(1) $$