Abstract Algebra: Theory and Applications · Problem 8.6.25

Isomorphisms — Thomas Judson

Mathematics — Abstract Algebra · dual-orthogonal-code

Problem

Let

C

be an

(n,k)

-linear code. Define the dual or orthogonal code of

C

to be

C^\perp = \{\mathbf{x} \in \mathbb{Z}_2^n : \mathbf{x} \cdot \mathbf{y} = 0 \text{ for all } \mathbf{y} \in C\}.

((a)) Find the dual code of the linear code

C

where

C

is given by the matrix

\begin{pmatrix} 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \end{pmatrix}.

((b)) Show that

C^\perp

is an

(n, n-k)

-linear code. ((c)) Find the standard generator and parity-check matrices of

C

and

C^\perp

. What happens in general? Prove your conjecture.