Lemmas

Question 1

Properties of the Hammin Distance

For any words x, y, z ∈ Fⁿ,

Answer 1

d(x,y) ≥ 0; d(x,y) = 0 iff x=y

d(x,y) = d(y,x)

d(x,z) ≤ d(x,y) + d(y,z) (triangle inequality)

Question 2

The number of points in a Hamming sphere

|S_r(y)| = ?

Answer 2

S_r​(y) = Σ₀^r (nCri) (q-1)ⁱ

Question 3

distance and weight

For any vectors v,y ∈ F_qⁿ, how are distance and weight related?

Answer 3

d(v,y) = w(v-y)

Question 4

If G is a generator matrix of C then C perp = ?

Answer 4

C perp = {v ∈ F_qⁿ: vG^T = 0}

Question 5

Each element of R_n is represented by how my polynomials of degree < n?

Answer 5

Exactly one polynomial of degree < n

Question 6

What is the length of dim of R(r,m)?

Answer 6

R(r,m) has length 2^m and dimension (mCr0) + (mCr1) + … + (mCrr)