WEBVTT 00:00:00.880 --> 00:00:05.710 Is there any chance we can not only detect a single-bit error but also correct the error 00:00:05.710 --> 00:00:07.770 to recover the original data? 00:00:07.770 --> 00:00:08.770 Sure! 00:00:08.770 --> 00:00:10.059 Here's how. 00:00:10.059 --> 00:00:15.190 By increasing the Hamming distance between valid code words to 3, we guarantee that the 00:00:15.190 --> 00:00:19.150 sets of code words produced by single-bit errors don't overlap. 00:00:19.150 --> 00:00:28.189 The set of code words produced by corrupting 000 (100, 010, or 001) has no code words in 00:00:28.189 --> 00:00:37.340 common with the set of code words produced by corrupting 111 (110, 101, or 011). 00:00:37.340 --> 00:00:41.660 Assuming that at most one error occurred, we can deduce the original code word from 00:00:41.660 --> 00:00:43.600 whatever code word we receive. 00:00:43.600 --> 00:00:50.490 For example, if we receive 001, we deduce that the original code word was 000 and there 00:00:50.490 --> 00:00:53.190 has been a single-bit error. 00:00:53.190 --> 00:00:58.030 Again we can generalize this insight: if we want to correct up to E errors, the minimum 00:00:58.030 --> 00:01:03.110 Hamming distance between valid code words must be at least 2E + 1. 00:01:03.110 --> 00:01:07.569 For example, to correct single-bit errors we need valid code words with a minimum Hamming 00:01:07.569 --> 00:01:09.590 distance of 3. 00:01:09.590 --> 00:01:14.130 Coding theory is a research area devoted to developing algorithms to generate code words 00:01:14.130 --> 00:01:17.799 that have the necessary error detection and correction properties. 00:01:17.799 --> 00:01:20.689 You can take entire courses on this topic! 00:01:20.689 --> 00:01:25.700 But we'll stop here with our basic insights: by choosing code words far enough apart (as 00:01:25.700 --> 00:01:30.530 measured by Hamming distance) we can ensure that we can detect and even correct errors 00:01:30.530 --> 00:01:33.039 that have corrupted our encoded data. 00:01:33.039 --> 00:01:33.530 Pretty neat!