If two digits were corrupted, we would end up misinterpreting the answer, but it is certainly better than nothing. If we received the message “101,” we could assume one of the digits had been corrupted in transit but still deduce that the sender had intended to say “yes.” Of course, this code is not perfect. For example, if we wanted to transmit only the messages “yes” and “no,” we could decide to encode “yes” as 111 and “no” as 000. One way to do this is to choose only a finite number of possible messages to transmit and build in some redundancy. You’re never saying “B as in bad,” which might leave the other person confused: “D as in dad? P as in pad?”Īn error-correcting code is, like the phonetic alphabet, a way to encode data that allows people to reconstruct the message even if the data has been altered in transit. If you have to spell your name over the phone, it’s easy for the person on the other end to mistake a B for a P, so it’s helpful to be able to say “B as in bravo.” The words that represent the letters in the phonetic alphabet were chosen so that even with a lot of static, there isn’t any ambiguity about which letter you mean. Take the NATO phonetic alphabet, better recognized by its first three words, alpha-bravo-charlie. The idea is to find a way of communicating that will be robust to small changes in the data that is transmitted. In the real world, think cell phones or fiber-optic cables, or even just trying to have a conversation in a noisy room. Here’s the situation: we want to transmit messages over a channel where data might get corrupted in transit. One way they show up quite naturally is in sewing, which I wrote about earlier this month. In my article about Viazovska’s sphere-packing breakthrough, I mentioned another application of high-dimensional sphere packing: data transmission and error-correcting codes. Surprisingly enough, though, they can arise quite naturally. In two dimensions, the sphere is a circle, and in three it is what we usually think of as a sphere.īecause high-dimensional spaces are difficult, if not impossible, for us to visualize, it’s easy to assume they don’t have any real-world significance. In any dimension, a sphere is the set of points that are all the same distance from one center point. There’s no limit to the number of dimensions we can study just by adding more coordinates. Just as three-dimensional space can be labeled using three coordinates-length, width, and height, or (x,y,z)-eight-dimensional space uses eight coordinates. It can be baffling to try to think about high-dimensional space, but mathematicians have a few tricks up their sleeves for it. Earlier this year, Maryna Viazovska showed how spheres can be most efficiently arranged in eight-dimensional space.
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