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 rank of the bar product is the sum of the two ranks:
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 (a) twice the weight of C1, and (b) the weight of C2:
For all ,
which has weight . Equally
for all and has weight . So minimising over we have
Now let and , not both zero. If then: