Probability of correcting errors

Error correcting codes are most simply described in terms of the errors they can certainly correct. For example, the Hadamard code used for the Mariner 9 probe to Mars encoded each 6-bit pixel to a 32-bit codeword in such a way that the original pixel could be recovered if no more than 7 bits were corrupted in transit. What is the probability that a pixel could be repaired if corrupted? That depends on your probability model. We will assume that the probability of each bit being flipped is p and that errors are independent. (Are errors independent, i.e. if a bit flips, is the next bit more or less likely to flip? That would depend on context.) It’s straight-forward to calculate the probability that 7 or fewer or fewer bits out of 32 flip; this is the cumulative distribution of a binomial random variable. The following Python code will return the probability of k or fewer successes out of n trials, each with probability of success p: from scipy.stats import binom print(binom.cdf(k, n,…