Bar product

In information theory, the bar product of two linear codes C₂ Ã¢ÂŠÂ† C₁ is defined as

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:

Proof

Let be a basis for and let be a basis for . Then the set

is a basis for the bar product .

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:

Proof

For all ,

which has weight . Equally

for all and has weight . So minimising over we have

Now let and , not both zero. If then:

If then

so

See also

• Reed–Muller code

References