Bar product

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&thinsp;(d, r) code in terms of the Reed–Muller codes RM&thinsp;(d&thinsp;&minus;&thinsp;1, r) and RM&thinsp;(d&thinsp;&minus;&thinsp;1, r&thinsp;&minus;&thinsp;1).

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

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


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