defs 8

Question 1

Boolean function

Answer 1

Denote by V^m the set of all binary words of length
m. (It is the same as F^m_2 but viewed without any vector space structure). A Boolean function is a (set-theoretical) function f : V^m → F_2.

Question 2

Boolean algebra on V^m

Answer 2

The ring/vector space of Boolean functions on V^m is called the Boolean algebra on V^m

Question 3

coordinate function

Answer 3

The Boolean function vi: V^m → F2 defined by
vi(x1, x2, . . . , xm) = xi
is called the ith coordinate function

Question 4

monomial function (or monomial)

Answer 4

To each subset {i1, . . . , ir} ⊆ {1, . . . , m} there corresponds the monomial function (or monomial)
vi1. . . vir of degree r

Question 5

rth order Reed-Muller code on V^m

Answer 5

The rth order Reed-Muller code on V^m, denoted R(r, m), is the subspace of the Boolean algebra on V^m spanned by monomials of degree at most r

(Here 0 ≤ r ≤ m.)

Question 6

bar product

Answer 6

Let C1, C2 ⊆ F^n_q be two linear codes. The linear code
|C1|C2| = {[ u | u + v ] : u ∈ C1, v ∈ C2}.

of length 2n is called the bar product of C1 and C2