Error-Control Block Codes for Communications Engineers
An engineer s guide to the theory behind the common random-error-control block codes, and their applications in digital communications.
TABLE OF CONTENTS Error-Control Block Codes for Communications Engineers
3.3: Relationship of Minimum Distance to Error Detection and Correction
3.3 Relationship of Minimum Distance to Error Detection and Correction
From Definition 3.4, the minimum Hamming distance of a code is the smallest Hamming distance between distinct codewords. For a linear block code, the all-zero vector is a codeword. Clearly, the minimum Hamming distance is equal to the minimum weight of its nonzero codeword, denoted as w min{ V}.
Given the parity-check matrix H of an ( n, k) linear code, one can determine the minimum distance of the code using the following theorem.
Theorem 3.1
For an ( n, k) linear block code, the minimum weight of a linear code is equal to the smallest number of columns of H that sums to zero.
Proof. VH T = v 0[ h 0,0 h 1,0 . . . h n ? k ?1,0] + v 1[ h 0,1 h 1,1 . . . h n ? k ?1,1] + . . . + v n ?1[ h 0, n ?1 h 1, n ?1 . . . h n ? k ?1, n ?1] = 0. The code symbol v j is associated with the vector [ h 0, j h 1, j . . . h n ? k ?1, j], for 0 ? j ? n ? 1, and [ h 0, j h 1,
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