Two involved lab projects will be assigned during the course. Each will require a write-up of around 5 pages. A project proposal of around 3 pages will be due almost 3 weeks before the final projects are due. The products of final projects will be oral and written reports. The written reports will be around 20 pages, and will be handed in twice — once about a month before the end of the class at which time revisions will be suggested, and again right before the end of the semester.
Students may work in groups of 2 or 3 on their projects, but must work individually on their written reports. This project will probably involve original research. Suggestions for topics will be provided.
Course Home. Lecture Notes. Download this Course. Text Fan, John L.
Description The field of Error-Correcting Codes has been revolutionized in the last decade by the emergence of iterative decoding techniques. We are talking about the set of all points within one unit of the first. In 2D, that's the inside of a circle. In 3D, that's the inside of a sphere. In 7D, it's--well, let's still call it a sphere. So here's the idea: a noisy channel typically displaces input words "points".
If it is most likely to displace them into spheres of limited radius about the original words, then we can think of each word that comes out as being some random point within a sphere. We just have to find the center of that sphere. That's hard to do, unless we restrict the words going into the channel. Let's make them as far apart in Hamming distance as possible.
For instance, if we use words that are no closer than three units to each other, then their unit spheres will not overlap. If the channel randomly displaces one of them by one unit, it still lies in the correct unit sphere. In principle, we can find the center of that sphere.
That tells us exactly what the original word was and lets us correct the error.
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Multiple errors can be corrected by placing all our input words at distances of five, seven, Mathematically, it's the same problem as packing marbles efficiently in a box. Wait, you're going to say, but don't we have to use all the words for our messages?
A Short Course in Information Theory
The answer is no. It works like any other substitution code. For example, it is possible to find 16 words in the 7D space that all lie at least three units from each other. So what we will do is take our original message, break it into blocks of four bits--of which there are exactly 16 possibilities--and assign each one of those 16 blocks to one of the 16 words in 7D space. In effect, we have added three bits of redundancy to our message by encoding every four bits into words that are seven bits long. But we can now actually correct any single error that might occur in any four-bit block of the original message.
Every possible point is accounted for and none is left out. The task of coding theory is to find good sphere packings in these spaces and to work out efficient algorithms for encoding and decoding the messages.
Claude Shannon, in his seminal work in this subject a half century ago, proved that finding good sphere packings is always possible. The idea of his proof is startling: he showed that if you use a space of sufficiently high dimension, then a set of words sphere centers taken at random is likely to work! Unfortunately, Shannon's Theorem does not provide efficient encoding or decoding methods. Clever people have found many ways to produce efficient error-correcting codes.
Usually they have exploited mathematical properties of these multi-dimensional spaces arising from the fact we can add vectors and, in many cases, multiply them.
Error-correcting codes (Information theory): Books
The multiplication is not what you might think it is, and is not necessarily simple: it's related to deeper properties of these spaces that enable us to think of them as a generalization of numbers, or finite fields. To illustrate what an error-correcting code ECC can do, I decided to implement a family of codes, the Hamming codes , which can correct one error per word, because these are exceedingly simple to encode and decode and they correspond to perfect sphere packings.
The smallest interesting Hamming code is a 7, 4 linear code. This means:. This business about addition is a seemingly minor detail, and you can safely ignore it, but it is a key property to making the encoding and decoding very simple. I have visually broken each code word into a sub-block of four bits followed by three more bits. Evidently, each sub-block is just the original message word.
We can think of the additional three bits as being check bits. They are generated by simple formulas:. Remember, when we sum bits, we are thinking of 1's as "odd" and 0's as "even. Formulas like this make it simple and quick to encode each block of four message bits. Technically, such formulas are maintained in data structures call "generator matrices.
For decoding purposes, instead of one checksum, there are three. In any valid encoded word,. Technically, such checksums are maintained in data structures called "parity-check matrices. Now suppose one or more of these checksums is one. Surely an error has occurred. General working knowledge of programming e.
From error correcting codes to inapproximability via the PCP Theorem
Familiarity with ASIC design e. Familiarity with machine learning techniques and tools would be beneficial.
Good communication and organization skills, ability to work in a team, positive and proactive problem-solving attitude. Excellent English language skills writing and presenting. Conditions of employment We offer a fixed-term 4 year challenging position for 4 years in a dynamic and ambitious university and a stimulating research environment.
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Related A Short Course on Error Correcting Codes
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