Coding Theory Calculator

Definition - Hamming Code

A Hamming code of order $r$ over $F = GF(q)$ is an $(n,k)$ -code over $F$ with $n = \ rac{q^r - 1}{q - 1}$ and $k = n - r$ , and with PCM $H_r$ , an $r \ imes n$ matrix whose columns are scalar multiples of each other.

Definition - Error Vector

Suppose $c \in C$ is sent and $r \in V_n(F)$ is received. The error vector is $e = r - c$ .

Algorithm - Decoding Algorithm for Single-Error Correcting Codes (Hamming Codes)

Input: PCM $H$ and a received word $r \in V_n(F)$ .

1:
Compute $s = Hr^T$
2:
If $s = 0$ , then accept $r$ as the transmitted word (so $e = 0$ ); STOP
3:
Compare $s$ with the columns of $H$ . If $s = \alpha h_i$ for some $i$ , then set $e = (0, \ldots, 0, \alpha, 0, \ldots, 0)$ ( $\alpha$ at $i$ th position), and decode to $c = r - e$ ; STOP
4:
Report that more than one error has occurred

Coding Theory Calculator

Decoding Hamming Codes

Definition - Hamming Code

Definition - Error Vector

Algorithm - Decoding Algorithm for Single-Error Correcting Codes (Hamming Codes)

Decoded Word

Modulo

PCM H

Received Word r