Bar product

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

[math]\displaystyle{ C_1 \mid C_2 = \{ (c_1\mid c_1+c_2) : c_1 \in C_1, c_2 \in C_2 \}, }[/math]

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:

[math]\displaystyle{ \operatorname{rank}(C_1\mid C_2) = \operatorname{rank}(C_1) + \operatorname{rank}(C_2)\, }[/math]

Proof

Let [math]\displaystyle{ \{ x_1, \ldots , x_k \} }[/math] be a basis for [math]\displaystyle{ C_1 }[/math] and let [math]\displaystyle{ \{ y_1, \ldots , y_l \} }[/math] be a basis for [math]\displaystyle{ C_2 }[/math]. Then the set

[math]\displaystyle{ \{ (x_i\mid x_i) \mid 1\leq i \leq k \} \cup \{ (0\mid y_j) \mid 1\leq j \leq l \} }[/math]

is a basis for the bar product [math]\displaystyle{ C_1\mid C_2 }[/math].

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:

[math]\displaystyle{ w(C_1\mid C_2) = \min \{ 2w(C_1) , w(C_2) \}. \, }[/math]

Proof

For all [math]\displaystyle{ c_1 \in C_1 }[/math],

[math]\displaystyle{ (c_1\mid c_1 + 0 ) \in C_1\mid C_2 }[/math]

which has weight [math]\displaystyle{ 2w(c_1) }[/math]. Equally

[math]\displaystyle{ (0\mid c_2) \in C_1\mid C_2 }[/math]

for all [math]\displaystyle{ c_2 \in C_2 }[/math] and has weight [math]\displaystyle{ w(c_2) }[/math]. So minimising over [math]\displaystyle{ c_1 \in C_1, c_2 \in C_2 }[/math] we have

[math]\displaystyle{ w(C_1\mid C_2) \leq \min \{ 2w(C_1) , w(C_2) \} }[/math]

Now let [math]\displaystyle{ c_1 \in C_1 }[/math] and [math]\displaystyle{ c_2 \in C_2 }[/math], not both zero. If [math]\displaystyle{ c_2\not=0 }[/math] then:

[math]\displaystyle{ \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} }[/math]

If [math]\displaystyle{ c_2=0 }[/math] then

[math]\displaystyle{ \begin{align} w(c_1\mid c_1+c_2) & = 2w(c_1) \\ & \geq 2w(C_1) \end{align} }[/math]

so

[math]\displaystyle{ w(C_1\mid C_2) \geq \min \{ 2w(C_1) , w(C_2) \} }[/math]

See also

• Reed–Muller code