Hadamard matrices are very special matrices, so you’d expect that they could be determined by less data than the full matrix. How much less? How many bits would it take to determine an n by n Hadamard matrix? More here.
As mentioned earlier, the Mariner 9 probe used an error correcting code based on Hadamard matrices. Each pixel in a Mariner photo was a 6-bit grayscale value, transmitted as a 32-bit codeword. This scheme could correct up to 7 flipped bits. What was the probability of no more than 7 bits being flipped? Given an overall error rate, how would you solve for the tolerable error per pixel? More here.
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