Bar product

In information theory, the bar product of two linear codes C2 ⊆ 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