Abstract Algebra: Theory and Applications · Problem 8.6.15

Isomorphisms — Thomas Judson

Mathematics — Abstract Algebra · coset-decoding

Problem

For each of the following matrices, find the cosets of the corresponding code

C

. Give a decoding table for each code if possible. ((a))

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

((b))

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

((c))

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

((d))

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

Hint. (a)

C

(10000)+C

(01000)+C

(00100)+C

(00010)+C

(11000)+C

(01100)+C

(01010)+C

. A decoding table does not exist for

C

since this is only a single error-detecting code.