Abstract Algebra: Theory and Applications · Problem 22.4.29

Galois Theory — Thomas Judson

Mathematics — Abstract Algebra · finite-fields-generator-parity-check

Problem

Let

C = \langle g(t) \rangle

be a cyclic code in

R_n

and suppose that

x^n - 1 = g(x)h(x)

, where

g(x) = g_0 + g_1 x + \cdots + g_{n-k}x^{n-k}

and

h(x) = h_0 + h_1 x + \cdots + h_k x^k

. Define

G

to be the

n \times k

matrix

G = \begin{pmatrix} g_0 & 0 & \cdots & 0 \\ g_1 & g_0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ g_{n-k} & g_{n-k-1} & \cdots & g_0 \\ 0 & g_{n-k} & \cdots & g_1 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & g_{n-k} \end{pmatrix}

and

H

to be the

(n-k) \times n

matrix

H = \begin{pmatrix} 0 & \cdots & 0 & 0 & h_k & \cdots & h_0 \\ 0 & \cdots & 0 & h_k & \cdots & h_0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ h_k & \cdots & h_0 & 0 & 0 & \cdots & 0 \end{pmatrix}.

(a) Prove that

G

is a generator matrix for

C

. (b) Prove that

H

is a parity-check matrix for

C

. (c) Show that

HG = 0