Bar product
In information theory, the bar product of two linear codes C2 ⊆ C1 is defined as
where 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 code in terms of the Reed–Muller codes RM and RM .
The bar product is also referred to as the | u | u+v | construction
or 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.The Hamming weight w of the bar product is the lesser of twice the weight of C1, and 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
