Unformatted text preview: mn parity bits C in
column order. The length of w is n = rc + r + c. This code is linear because all the parity
bits are linear functions of the message bits. The rate of the code is rc/(rc + r + c).
We now prove that the rectangular parity code can correct all single-bit errors.
Proof of single-error correction property: This rectangular code is an SEC code for all
values of r and c. We will show that it can correct all single bit errors by showing that
its minimum Hamming distance is 3 (i.e., the Hamming distance between any two code
words is at least 3). Consider two different uncoded messages, Mi and Mj . There are three
cases to discuss:
• If Mi and Mj differ by a single bit, then the row and column parity calculations
involving that bit will result in different values. Thus, the corresponding code words,
wi and wj , will differ by three bits: the different data bit, the different row parity bit,
and the different column parity bit. So in this case HD(wi , wj ) = 3.
• If Mi and Mj differ by t...
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This document was uploaded on 02/26/2014 for the course CS 6.02 at MIT.
- Fall '13