3.5.1 Repetition Codes

Repetition is the simplest form of error protection. Each information digit may be transmitted n times. The generator and parity-check matrices of an ( n, 1) binary repetition block code are

(3.21)

and

(3.22)

respectively. It can be seen that the minimum Hamming distance of an ( n, 1) repetition block code is n. In general, n transmissions of the same digit enable n ? 1 errors to be detected, or ?( n ? 1)/2 errors to be corrected.

3.5.2 Single-Parity-Check Codes

A single-parity-check code of block length n may be formed by taking a parity-check over k = n ? 1 information digits. The parity-check digit v _{n ?1} may be written as

(3.23)

where + implies modulo-2 addition. We chose an even-parity rule so that each codeword has an even number of ones. The generator and parity-check matrices of an ( n, n ? 1) binary single-parity-check block code are

(3.24)

and

(3.25)

respectively. Single, triple, and all odd number of errors in the block n