User:ECCclass/Elias-Bassalygo

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

Let $$B_q(\boldsymbol{y}, \rho n) =\{ \boldsymbol{x} \in [q]^n | \Delta(\boldsymbol{x}, \boldsymbol{y}) \le \rho n \}$$ be the Hamming Ball of radius $$ \rho n $$ centered at $$\boldsymbol{y}$$. Let $$ Vol_q(\boldsymbol{y}, \rho n) = |B_q(\boldsymbol{y}, \rho n)| $$ 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. irrelevant with position of \boldsymbol{y}. In particular, $$B_q(\boldsymbol{y}, \rho n) =B_q(\boldsymbol{0}, \rho n) $$

With large enough $$n$$, the rate $$R$$ and the relative distance$$\delta$$ satisfies the Elias-Bassalygo bound: $$R \le 1 - H_q( J_q(\delta))+o(1) $$

where


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

is the q-ary entropy function and


 * $$J_q(\delta) \equiv_\text{def} (1-{1\over q})(1-\sqrt{1-{q \delta \over{q-1}}} $$ is a function related with Johnson bound

To prove the Elias–Bassalygo bound, we'll need the following Lemma:

Lemma 1: Given a q-ary code, $$C\subseteq [q]^n $$ and $$ 0\le  e\le  n$$, there exists a Hamming ball of radius $$e$$ with at least $${|C|Vol_q(0,e)} \over {q^n}$$ codewords in it.

Proof of Lemma 1: We prove Lemma 1 using probability method. Let's random pick a received word $$y \in [q]^n$$. The expected size of overlapped region between $$ C$$ and the Hamming ball centered at $$y$$ with radius $$e$$, $$|B_q(y,e) \cap C|$$ is $$Vol_q(y,e) {{|C|} \over {q^n}}$$ since $$y$$ is (uniform) randomly selected. Since this is the expected value of the size, there must exist at least one $$y$$ such that $$|B_q(y,e) \cap C| \ge Vol_q(y,e) {{|C|} \over {q^n}} = {{|C|Vol_q(0,e)} \over {q^n}}$$, 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 1, there exists a Hamming ball with $$B$$ codewords such that $$B\ge { {|C|Vol(0,e)} \over {q^n}} $$

By Johnson Bound Johnson bound, we have $$B\le qdn$$. Thus,

$$\mid C \mid \le qnd \cdot {{q^n} \over {Vol_q(0,e)}} \le q^{n(1-H_q(J_q(\delta))+o(1))}$$

The second inequality follows from lower bound on the volume of a Hamming ball: $$ Vol_q(0, \lfloor {{d-1} \over 2} \rfloor) \le q^{H_q({\delta \over 2})n-o(n)} $$

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

Therefore we have


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