Example: A one-error correcting code

• The following code takes four bits and forms a code word of 7 bits.
• A code that takes a fixed length block of bits and produces a fixed length code word is called a block code.
• For long messages, the output of the source coder is broken up into blocks of 4 bits.
• Each block is coded into 7-bit code words.
• The ratio of input length to coded length is called the rate of the code.
• Hence, the code below is a rate $\frac{4}{7}$ code.
• Let us denote the four input bits as b(1), b(2), b(3), and b(4).
• Similarly, the seven output (coded) bits are denoted c(1) through c(7).
• Our coder forms output bits from the input bits as follows:
  
  $$\begin{array}{cl} c(1) & = b(1)\\ c(2) & = b(2)\\ c(3) &... ...us b(4)\\ c(7) & = b(1) \oplus b(2) \oplus b(4) \end{array}$$