Gibbs' inequality

In information theory, the bar product of two linear codes C₂ ⊆ C₁ is defined as

$C_{1}\mid C_{2}=\{(c_{1}\mid c_{1}+c_{2}):c_{1}\in C_{1},c_{2}\in C_{2}\},$

where (a | b) denotes the concatenation of a and b. If the code words in C₁ are of length n, then the code words in C₁ | C₂ are of length 2n.

The bar product is an especially convenient way of expressing the Reed–Muller RM (d, r) code in terms of the Reed–Muller codes RM (d − 1, r) and RM (d − 1, r − 1).

The bar product is also referred to as the | u | u+v | construction or (u | u + v) construction.

Properties

Rank

The rank of the bar product is the sum of the two ranks:

$\operatorname {rank} (C_{1}\mid C_{2})=\operatorname {rank} (C_{1})+\operatorname {rank} (C_{2})\,$

Proof

Let $\{x_{1},\ldots ,x_{k}\}$ be a basis for $C_{1}$ and let $\{y_{1},\ldots ,y_{l}\}$ be a basis for $C_{2}$. Then the set

$\{(x_{i}\mid x_{i})\mid 1\leq i\leq k\}\cup \{(0\mid y_{j})\mid 1\leq j\leq l\}$

is a basis for the bar product $C_{1}\mid C_{2}$.

Hamming weight

The Hamming weight w of the bar product is the lesser of (a) twice the weight of C₁, and (b) the weight of C₂:

$w(C_{1}\mid C_{2})=\min\{2w(C_{1}),w(C_{2})\}.\,$

Proof

For all $c_{1}\in C_{1}$,

$(c_{1}\mid c_{1}+0)\in C_{1}\mid C_{2}$

which has weight $2w(c_{1})$. Equally

$(0\mid c_{2})\in C_{1}\mid C_{2}$

for all $c_{2}\in C_{2}$ and has weight $w(c_{2})$. So minimising over $c_{1}\in C_{1},c_{2}\in C_{2}$ we have

$w(C_{1}\mid C_{2})\leq \min\{2w(C_{1}),w(C_{2})\}$

Now let $c_{1}\in C_{1}$ and $c_{2}\in C_{2}$, not both zero. If $c_{2}\not =0$ then:

${\begin{aligned}w(c_{1}\mid c_{1}+c_{2})&=w(c_{1})+w(c_{1}+c_{2})\\&\geq w(c_{1}+c_{1}+c_{2})\\&=w(c_{2})\\&\geq w(C_{2})\end{aligned}}$

If $c_{2}=0$ then

${\begin{aligned}w(c_{1}\mid c_{1}+c_{2})&=2w(c_{1})\\&\geq 2w(C_{1})\end{aligned}}$

so

$w(C_{1}\mid C_{2})\geq \min\{2w(C_{1}),w(C_{2})\}$

See also

• Reed–Muller code