defs 6

Question 1

Distance Theorem for Linear Codes

Answer 1

Let C ⊆ F^n_q be a linear code with check matrix H. Then d(C) = d if and only if every set of d−1 columns of H is linearly independent and some set of d columns of H is linearly dependent.

Question 2

line

Answer 2

A line is a 1-dimensional subspace of the vector space F^n_q

Question 3

representative vector

Answer 3

A representative vector of a line is a non-zero vector u from that line. The line is then given by {λu | λ ∈ F_q}.

Question 4

projective space

Answer 4

The projective space P_n−1(F_q) is the set of all lines in F^n

Question 5

Hamming codes

Answer 5

Let r ≥ 2 be given. We let Ham(r, q) denote an Fq-linear code whose check matrix has columns which are representatives of the lines in P_r−1(Fq), exactly one representative vector from each line

Question 6

Ham(r, q) is a perfect…

Answer 6

Ham(r, q) is a perfect [n, k, d]q code where
n =(q^r − 1) / q − 1

k = n − r, d = 3.

Question 7

Decoding algorithm for a Hamming code

Answer 7

Let a Hamming code be given by its check matrix H. Suppose that a vector y is received.
• Calculate S(y) = yHT
• Find a column of H such that S(y) = λ × that column. (ith column)
• Subtract λ from the ith position in y. The result is the codevector DECODE(y).

Question 8

A simplex code

Answer 8

A simplex code Σ(r, q) is defined as Ham(r, q)⊥

Question 9

weight enumerator of the simplex code

Answer 9

W_Σ(r,q) (x, y) = x^n + (q^r − 1)x^[n−q^r−1]y^[q^r−1]

where n = q^r − 1 /q − 1