defs 4

Question 1

linear code

Answer 1

A linear code is a subspace of the vector space F^n_q

Question 2

weight of vector

Answer 2

The weight w(v) of a vector v ∈ F^n_q is the number of non-zero symbols in v.

Question 3

weight of code

Answer 3

The weight w(C) of the code C ⊆ F^n_q is w(C) = min{w(v) | v ∈ C \ {0}}

Question 4

zero sum code

Answer 4

Z = {(v1, v2, . . . , vn) ∈ F^n_q | v1 + v2 + · · · + vn = 0 in Fq}

Question 5

binary even weight code

Answer 5

En = {v ∈ F^n_2 : w(v) is even}

Question 6

generator matrix

Answer 6

Let C ⊆ F^n_q be a linear code. A generator matrix of
C is a matrix G =
[r1]
[r2]
[.  ]
[.  ]
[.  ]
[rk]

where the row vectors r1, . . . , rk are a basis of C

Question 7

• the rows of G are…..
• the number n of columns of G is the …..
• the number k of rows of G is …..
• the number of codevectors is ….
• the dimension of the code is …..

Answer 7

• the rows of G are linearly independent
• the number n of columns of G is the length of the code
• the number k of rows of G is the dimension, dim(C), of the code
• the number of codevectors is M = q^k;
• the dimension of the code is equal to its information dimension: k = log_q M

Question 8

generator matrix G is in standard form

Answer 8

A generator matrix G is in standard form if its leftmost colums form an identity matrix:
G = [Ik | A]
.
Note that the entries in the last n − k columns, denoted by ∗, are arbitrary elements of F_q

Question 9

If a generator matrix in standard form exists for a linear code C, it is unique, and any generator matrix can be brought to the standard from by the following operations:

Answer 9

(R1) Permutation of rows.
(R2) Multiplication of a row by a non-zero scalar.
(R3) Adding a scalar multiple of one row to another row

Question 10

coset of y

Answer 10

Given a linear code C ⊆ F^n_q and a vector y ∈ F^n_q
the set
y + C = {y + c | c ∈ C}
is called the coset of y

Question 11

coset leader

Answer 11

A coset leader of a coset y + C is a vector of minimum

weight in y + C

Question 12

the standard array decoder

Answer 12

Preparation: Construct a standard array for C.

Decoding:
• Receive a vector y ∈ F^n_q
• Look up y in the standard array:
– The row of y starts with its chosen coset leader ai
.
– The column of y starts with y − ai
.
• Return the topmost vector of the column of y as DECODE(y).

Question 13

weight enumerator

Answer 13

The weight enumerator of a linear code C ⊆ F^n_q is

WC(x, y) = SUM[v∈C] x^(n−w(v)) y^w(v)

= A0x^n + A1x^(n−1)y + A2x^(n−2)y^2 + . . . + Any^n

where Ai = #{v ∈ C : w(v) = i}. The weight enumerator of C is a polynomial in two variables x, y