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Error Correction

Module by: Don Johnson

Summary: So-called linear codes create error-correction bits by combining the data bits linearly. Topics discussed include generator matrices and the Hamming distance.

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So-called linear codes create error-correction bits by combining the data bits linearly. The phrase "linear combination" means here single-bit binary arithmetic.

Table 1
0⊕0=0 0 0 0 1⊕1=0 1 1 0 0⊕1=1 0 1 1 1⊕0=1 1 0 1
0∧0=0 0 0 0 1∧1=1 1 1 1 0∧1=0 0 1 0 1∧0=0 1 0 0

For example, let's consider the specific (3,1) error correction code described by the following coding table and, more concisely, by the succeeding matrix expression. c1=b1 c 1 b 1 c2=b1 c 2 b 1 c3=b1 c 3 b 1 or c=Gb c G b G= 111 G 1 1 1 c= c1c2c3 c c 1 c 2 c 3 b= b1 b b 1

The length-KK block of data bits is represented by the vector bb, and the length-NN output block of the channel coder, known as a codeword, by cc. The generator matrix GG defines all block-oriented linear channel coders.

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