Dynamic errors of numerical methods of ODE discretization

The Elias-Bassalygo bound is a mathematical limit used in coding theory for error correction during data transmission or communications. The properties of the Elias-Bassalygo bound are defined, below, using mathematical expressions.

Definition

Let $C$ be a $q$ -ary code of length $n$ , i.e. a subset of $[q]^{n}$ . (Each $q$ -ary block code of length $n$ is a subset of the strings of $𝒜_{q}^{n},$ where the alphabet set $𝒜_{q}$ has $q$ elements). Let $R$ be the rate of $C$ and $δ$ (delta) be the relative distance.

Let $B_{q} (y, ρ n) = {x \in [q]^{n} | Δ (x, y) \leq ρ n}$ be the Hamming ball of radius $ρ n$ centered at $y$ . Let $V o l_{q} (y, ρ n) = | B_{q} (y, ρ n) |$ be the volume of the Hamming ball of radius $ρ n$ . It is obvious that the volume of a Hamming Ball is translation-invariant, i.e. irrelevant with position of $y$ . In particular, $| B_{q} (y, ρ n) | = | B_{q} (0, ρ n) |$ .

With large enough $n$ , the rate $R$ and the relative distance $δ$ satisfies the Elias-Bassalygo bound: $R \leq 1 - H_{q} (J_{q} (δ)) + o (1),$

where

H_{q} (x) \equiv_{def} - x \cdot \log_{q} \frac{x}{q - 1} - (1 - x) \cdot \log_{q} (1 - x)

is the q-ary entropy function and

J_{q} (δ) \equiv_{def} (1 - \frac{1}{q}) (1 - \sqrt{1 - \frac{q δ}{q - 1}})

is a function related with Johnson bound.

Proof

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

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

Proof of Lemma 1: To prove Lemma 1, use the probability method. Randomly 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 $V o l_{q} (y, e) \frac{| C |}{q^{n}}$ since $y$ is (uniform) randomly selected. Since this is the expected value of the size, there must exist at least one $y$ such that $| B_{q} (y, e) \cap C | \geq V o l_{q} (y, e) \frac{| C |}{q^{n}} = \frac{| C | V o l_{q} (0, e)}{q^{n}}$ , otherwise the expectation must be smaller than this value.

Now we prove the Elias–Bassalygo bound.

Define $e = n J_{q} (δ) - 1$ .

By Lemma 1, there exists a Hamming ball with $B$ codewords such that $B \geq \frac{| C | V o l (0, e)}{q^{n}}$

By the Johnson bound, we have $B \leq q d n$ . Thus,

∣ C ∣ \leq q n d \cdot \frac{q^{n}}{V o l_{q} (0, e)} \leq q^{n (1 - H_{q} (J_{q} (δ)) + o (1))}

The second inequality follows from lower bound on the volume of a Hamming ball: $V o l_{q} (0, ⌊ \frac{d - 1}{2} ⌋) \leq q^{H_{q} (\frac{δ}{2}) n - o (n)}$ .

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

Therefore we have

R = \frac{\log_{q} | C |}{n} \leq 1 - H_{q} (J_{q} (δ)) + o (1)

References

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Dynamic errors of numerical methods of ODE discretization

Contents

Definition

Proof

See also

References

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Dynamic errors of numerical methods of ODE discretization

Definition

Proof

See also

References

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