Placeholder Image

Subtitles section Play video

  • We're going to talk about going from Galois fields to Reed-Solomon codes.

  • We must be mad, Sean. Really, I mean, so many of you have said: "Just do Reed-Solomon,

  • y' know. You've done Hamming codes. They [Reed-Solomon] can't be that much more complex?" Oh yes they can!

  • The stuff we have done earlier - and we've done a fair bit on Hamming codes, which if you

  • remember are basically going to correct a ... put right a single error. That all

  • happened in the 1960s. And in two of the videos out there, which we'll put the

  • links out for you, I did a thing called Multi-Dimensional Error Correction where

  • I had two bits of information (which was a San Francisco state of weather). So

  • there was two bits for the state of the weather, like I think it was '11' for "sunny",

  • or something like that. But in order to protect those bits against a

  • 1-bit damage, I had to add no fewer than 3 extra bits. I had to make it into

  • a 5-bit code. But it was OK, because checking up whether it had been damaged

  • was easy. It was a simple parity check. It was effectively saying : "Look, here's the

  • 3 parity-check bits at the end". I've made the rule that overall it must be

  • even parity. I know that one of them says '1' Another one says '0' but this [third] bit I

  • don't know. But it's got to be even parity overall so if I've only got a single

  • '1'and a '0', I need another '1' to make that grouping be even parity.

  • Now that's the easy error correction. How did it advance from that? What changed?

  • Well, we're talking about 10 years hence. I mean Richard Hamming was very definitely

  • late '50s / early '60s. The next phase of this or, this "big jump forward"

  • thst we would want to do, was by Reed and Solomon [and] was 10 years later. It was late '60s / early '70s.

  • O!h and big surprise: Were Reed and

  • Solomon at Bell Labs ?! No! For a change. Everybody else seemed to be at Bell Labs

  • [but] Reed and Solomon were, I think, at MIT's Lincoln Laboratory. But there

  • was a realization that if you wanted more powerful error correction there

  • were several things you could do but, the more they looked into it, the more they

  • found themselves having to learn about Pure Mathematics concepts called Galois

  • Fields. And this is finite field arithmetic where you've got to be able

  • to find a multiplicative and an additive inverse. We've tried to prepare you for

  • this by doing ISBN book codes, which is a very simple manifestation of those two

  • things. We thought in our early things [i.e. videos] where a code word became a string of bits.

  • And some of those are information bits. And interleaved with them, or parked at

  • the right-hand end very often, were parity check bits. What we're now going

  • to try and do is instead of just having one great long string of bits, linearly,

  • we're going to try and make it be almost two-dimensional. Think of every one of

  • these positions in your code-word now as not being a single bit but a column

  • of bits. Let's say it's an 8 bit column. If you're doing something like Reed-Solomon

  • correction in a CD context, y'know, how big is the bucket size of

  • these columns? I think it's something like 40-odd bits but then even that

  • couldn't cope. They had, like, two layers of Reed-Solomon encoding - one backing up

  • the other. So, if you like, the filling up of these things, instead of linearly

  • going like that - and then at the very far end you put a few parity check bits - what

  • we now do is we declare that every one of these positions isn't a *bit* position

  • it's a *symbol* position but a symbol can be multiple bits, OK? For the sake of

  • argument let's say it's an 8-bit symbol, a byte. The way they get filled up is the

  • bit stream comes in and it fills up a column of 8, and then it fills up a next

  • column 8 and a next column of 8. So it's almost like we've got a 2-dimensional

  • array of bits - and symbols in that direction-

  • but every symbol is composed of 'n' bits, as it goes along. What's the advantage of

  • doing that? Well, you can see one advantage, when you think about it,

  • straight away. Hamming codes, for example, the old way, tended to presume you've got

  • the occasional error, now and then, wide apart. What this is anticipating - if you

  • can fill up symbol positions - is you might get 'burst errors'. Yeah, you might

  • get - here we go - bits coming off a CD, trying to play your music.

  • It's encoded music. Now, if they're filling up a bucket in a column,

  • in some sense, and then moving on to the next one

  • there is just a chance that a localized scratch will get all its 'bit-clobbering'

  • over and done with, within two symbols shall we say. So, we know that we can

  • devise codes that will can detect and potentially correct a certain number of

  • errors but if we can make it so that those errors are not [only] bit errors within

  • the symbol but just the symbol itself ... something's wrong with it! Right?

  • You then might stand a chance if you've got, again, parity-check *symbols* at

  • the far right-hand end - not [just] parity-check bits, if they've got enough information

  • in them, you might be able to say: "Something went wrong. I got a burst error

  • there, as a scratch on that CD. Can I put it right?" Yes, but it's not going to be

  • simple-minded parity checking; it's gonna be serious hardcore stuff. Because those

  • check symbols at the end will normally be arranged so that if the information

  • is correct and nothing's got damaged there'll be zeros. If something gets

  • damaged, the first thing you know about it is that the parity-check symbols at

  • the end are nonzero. You're getting [perhaps] 3, 5, 12, 15. What does that mean?

  • well the answer is using lots and lots of detective work - by the

  • way those symbols that you put on the far right-hand end - revel in the name of

  • "syndrome", which i think is a wonderful word. And my first thought was what on

  • earth are pure mathematicians, or communications engineers, doing with

  • syndromes? They're medical aren' they?! Well, I looked up in the dictionary and as far

  • as I can make out it all makes sense. If you have a certain syndrome it means you

  • are exhibiting symptoms of an underlying problem

  • So the grouping of symptoms that's caused by an underlying problem is often

  • called a syndrome. [There's] something like the Guillain-Barre syndrome isn't there? I don't know what

  • it is but I'm no doubt you'll all put me right on that. But you can see it makes

  • sense. The signal that something has gone wrong is you get all of these

  • information bytes or even bigger columns, multiple bytes, whatever. But right at the

  • end is now a "checksum from hell". You have got a syndrome a set of remainders if

  • you like, that are not zeros. Given only that information, how can you work

  • backwards and find out which of these columns got hit and where in the column

  • it got hit? And the answer is: By using Galois Field theory over finite fields and

  • doing lots and lots of long divisions and additions. So, the bottom line is that

  • for this work and particularly if you're using, as we are of course now, powers of

  • two, Galois said: "I can liberate you from it having to be primes all the time,

  • I can do powers of primes. And for all of you future computer scientists that I'm

  • not even aware of, you'll love this because your beloved 2 comes into the real

  • world because where you say '4' and I say: Don't think of it as 2 x 2

  • Think of it as 2 ^ 2." And so he said: "Yeah, I can do powers of

  • any prime, including two, but what he didn't say is: Here are the rules

  • if you you light on 2 and say you want to use my methods, here's rule number 1.

  • Everything must be done modulo 2 Not modulo your bigger number - like 256 -

  • that's different. Modulo 2. So, for addition what do we know

  • about addition modulo 2, from Bletchley Park Sean? What do the

  • mathematical logicians call 'addition modulo 2' ? >> Sean: This is exclusive OR

  • >> DFB: It's Exclusive OR! So, no sweat for computer scientists. All our additions of these

  • numbers, represented in bytes or whatever, are going to be done

  • with Exclusive OR. Worse still - as we found out in the ISBN previous example -

  • you've got to be able to find multiplicative inverses. And that's going

  • to lead us into doing long divisions modulo 2 in a Galois Finite Field. Now

  • that is a bit hair-raising but not too terrifying. But if you're prepared in the

  • end to do all of that work then fine. You can use your 'syndrome' and analyze

  • it to tell you where the errors went. That's a lot of equation solving. For those of

  • you into these things it's like solving lots of simultaneous equations saying [for example]:

  • "The only logical solution to this damage is it absolutely must be column 3 and

  • column 13. That's where the damage has come. Does that syndrome always work back

  • to getting your precisely one answer? Yes - it's just magic the way that this

  • error correction works but it is complex because the brute force way to do it is

  • to get a set of simultaneous equations and - some of you will know this - get lots

  • of simultaneous equations to solve you use 'matrix inversion'. That is computationally

  • a very heavy undertaking. And so you're looking all the time for simplifications

  • because, if you don't, you're sitting there, like Sean and I were in the early

  • '80s, thinking: " ... this error correction [for CDs] this is in real time! How is this thing

  • solving sets of matrix equations in real time to make sure I can listen to this

  • CD without hearing the scratches? Where's our supercomputer free with every CD - a

  • Cray XMP [in those days] - to solve your syndrome equations? Nope! what's the answer? You shake

  • head and think: it's those hardware types! It's custom

  • hardware! Yes, custom hardware can fairly easily attain supercomputer capabilities

  • so long as it's in a tightly defined field. And it turns out, thank heavens,

  • that error correction in Reed-Solomon schemes lends itself very nicely indeed

  • to things that computer engineers love like "shift registers" [and pipelining] and all sorts of

  • hardware 'specials'.

We're going to talk about going from Galois fields to Reed-Solomon codes.

Subtitles and vocabulary

Operation of videos Adjust the video here to display the subtitles

B1 parity solomon reed galois correction column

Reed Solomon Encoding - Computerphile

  • 1 0
    林宜悉 posted on 2020/03/27
Video vocabulary