Bar product

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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 (dr) 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[1] or (u | u + v) construction.[2]

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{align} 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{align}

If $c_2=0$ then

\begin{align} w(c_1\mid c_1+c_2) & = 2w(c_1) \\ & \geq 2w(C_1) \end{align}

so

$w(C_1\mid C_2) \geq \min \{ 2w(C_1) , w(C_2) \}$