defs 7

Question 1

cyclic shift

cyclic code

Answer 1

For a vector a = (a0, a1, . . . , an−1) ∈ F^n_q, we denote s(a) = (an−1, a0, . . . , an−2) and call the vector s(a) the cyclic shift of a.

A cyclic code in F^n_q is a linear code C such that
a ∈ C ⇒ s(a) ∈ C.

Equivalently, a cyclic code is a linear code C such that s(C) = C

Question 2

ring

Answer 2

A (commutative) ring is an abelian group (R, +) equipped with an extra operation × (multiplication), such that for all a, b, c ∈ R (writing ab for a × b):

(ab)c = a(bc); ab = ba;
∃1 ∈ R: 1 =! 0, 1a = a;
a(b + c) = ab + ac

Question 3

algebra

Answer 3

An algebra A over the field Fq is a ring which is also a vector space over Fq, such that

∀λ ∈ Fq, a, b ∈ A, (λa)b = a(λb) = λ(ab).

That is, scaling one of the factors in a product ab by λ has the same effect as scaling the whole product

Question 4

algebra Rn

Answer 4

As a vector space, Rn consists of all polynomials of degree less than n in Fq[x]. Multiplication in Rn is defined by

f(x), g(x) ∈ Rn → remainder of f(x)g(x) when divided by x^n − 1

and is known as multiplication modulo x^n − 1.

Question 5

ideal of algebra R

Answer 5

An ideal of an algebra R is a subspace I ⊆ R such that RI⊆ I.

Question 6

generator polynomial of a cyclic code C

Answer 6

We say that g(x) ∈ Fq[x] is a generator polynomial of a cyclic code C if g(x) is monic, and C consists of all multiples of g(x) of degree less than n. We say that the generator polynomial of the zero code, {0}, is x^n − 1.

Question 7

check polynomial of C

Answer 7

Let g(x) be the generator polynomial of a cyclic code C. The polynomial h(x) such that g(x)h(x) = x^n − 1 is called the check polynomial of C. 
If deg g(x) = r, then deg h(x) = n − r, and h is monic.

Question 8

parameter equivalent

Answer 8

We say that two codes are parameter equivalent, if they both are [n, k, d]q-codes for some n, k, d and