Error Correction & International Book Codes - Computerphile

Moving on from crude error correction to more sophisticated methods, Professor Brailsford demostrates using the ISBN 'book code'.
Error Correction: • Error Correction - Com...
Reed Solomon Codes: COMING SOON
/ computerphile
/ computer_phile
This video was filmed and edited by Sean Riley.
Computer Science at the University of Nottingham: bit.ly/nottscomputer
Computerphile is a sister project to Brady Haran's Numberphile. More at www.bradyharan.com

[@michaelxu5698]

My father is a very proper mathematician and I've always lived in his dismissal of discrete mathematics. I'm now a CS guy, thanks for sharing the fascination and inspiration to not be dismissive!

[@gordonrichardson2972]

And the book title is ISBN 0201135027 - Theory of Information and Coding by Robert Mceliece.

[@sanferrera]

Thank you. I was also wondering.

[@johnfrancisdoe1563]

Gordon Richardson That's a famous name in the field. Never read his work, yet recognize his name instantly.

[@tamasdemjen4242]

Galois practically revolutionized mathematics while he was a teenager. He solved a polynomial root problem standing for 350 years, and laid the foundations for abstract algebra. Imagine what he could've done had he lived a few decades longer. His last words: Don't cry, Alfred! I need all my courage to die at twenty!

[@profdaveb6384]

Yes what an amazing talent he was. His brother and his friends had great problems getting his stuff published after his death - probably because hardly any mathematician in the world truly understood how amazing it all was. I think Joseph Liouville , in the end, published it all in his own journal. Hurrah!

[@noxabellus]

_Sees Professor Brailsford_ _Clicks without reading title_

[@cousindave1]

Same here!

[@sebastianelytron8450]

Second only to Dr. Pound in my book. Much better than Bagley though.

[@techtipsuk]

noxabellus usually I think moron when I read comments like this but with the Prof exceptions do have to be made.

[@sandeepvk]

what about Rebecca Tickle ?

[@srmutnuri]

Such a joy watching the prof speak so passionately!

[@stoneskull]

Professor Brailsford is like the computer Santa. So patient and good-humoured. terrific to listen to. Man, I love computerphile.

[@myselfremade]

"But we have computers, do we care?" Brailsford, 2019

[@aditya95sriram]

I am extremely intrigued by what Dr. Brailsford mentioned towards the end about Galois Theory. Has this been covered in some video already ? If not is it possible to make a video about the same ?

[@__mk_km__]

Amazing! I've never been able to tackle the Reed-Solomon EC, but now that video is a pretty solid background. Can't wait for the Reed-Solomon video!

[@DancingRain]

I'm looking forward to the video on Reed-Solomon codes. Cheers to you, and have a wonderful new year. :)

[@harrysvensson2610]

Will you ever cover CRC (CyclicRedundancy Check)? It would be interesting to hear how the polynomials work, because I've never understood what makes one polynomial better than another. Is Hadamard Codes coming too?

[@superscatboy]

"Prime numbers, as we know, are completely magic"

[@sonicthehedgehog5088]

What a coincidence, I haven't watched Computerphile in a while because I was busy trying to get Reed-Solomon codes to work in C#, after many failed attempts I pretty much succeeded this time, and guess what got uploaded? another video on error correction! _Sees end of the video and the description_ What?! after so many videos on error correction they finally are going to do it. Reed-Solomon codes!

[@jeremiahmullikin]

I am very fortunate to have viewed this.

[@MrHkrammes]

Great class! I only miss the last video on the series, as it seems not to be available on youtube yet :(

[@nab-rk4ob]

Little Feat! Spelled it right the first time. I laughed at the beginning when he was discussing picking the right number to do the test with. I do the same thing when trying to figure out how many characters my password length will be.

[@CathyInBlue]

Only ISBN-10 (10 digits) is modulo 11. ISBN-13 is modulo 10. A snag with modulo 11 arithmetic is your get not only 0-9, but 10. How do you represent 10 in a single (check) character? They chose the letter X. I suppose because it looks like the Roman numeral for 10. So, that is why when you are looking at a lot of 10 digit ISBNs, some of them end in the check digit of X. A 13 digit ISBN will never terminate in an X, because modulo 10 is sane in a decimal number system. Also, if the math of an ISBN doesn't work out properly, it doesn't actually give you any information as to WHICH digit (of 10 or 13) is wrong. It could even be the check digit itself that's wrong. It just tells you that there's SOMETHING wrong with those ISBN digits. Which one? No way of telling.

[@ferulebezel]

ISBN-13 is just what Americans call it. It's really just an EAN. to generate it take an ISBN, drop the check digit, prepend 978 (for the fictional country of Bookland), and calculate a new check digit.

[@CathyInBlue]

@@ferulebezel …by an algorithm completely different from the 10-digit ISBN. Every ISBN look up website I know of calls them ISBN-13s to distinguish from ISBN-10s, what with their completely different check digit algorithms. EAN is not synonymous with ISBN-13. All ISBN-13s are EANs, but not all EANs are ISBNs.

[@p_serdiuk]

@@CathyInBlue All EANs that have the prefixes 978 or 979-1 to 979-9 are ISBNs.

[@CathyInBlue]

@@p_serdiuk #NotAllEANs

[@carly09et]

Context can sort the forks out. The code space is finite but not complete so valid crashes are VERY rare :) < 10^-3.

[@SwissSareth]

Discrete Mathematics. First year undergraduate. Hated this stuff when I had to learn it back then. Now look at me getting excited about it because of Professor Brailsford...

[@seanmortazyt]

what a wonderful explanation!

[@DiamondSane]

@Computerphile Please check subtitles for "Busy Beaver Turing Machines"

[@thereadingcompany332]

Wonderful explanation, salute sir.

[@PrettyBlueThings]

Thank you Professor Brailsford! :)

[@JNCressey]

2:43 "I'm going to be talking about integers but including zero at the bottom end." Integers actually include the negative numbers also. The phrasing of 'zero at the bottom end' and omission of mentioning negatives makes it sound like the set being described is the natural numbers. However, the set of natural numbers fails even sooner in the following steps since it does not contain additive inverses.

[@Y2KMailliw]

Is the Reed Solomon Code video still coming soon?

[@Nadia1989]

This is a better example of operations in Zn that all those I had at class! Could it be possible to get subtitles?

[@sonicthehedgehog5088]

Is that Reed-Solomon video still coming? or does it only exist in the same dimension as that one Numberphile2 "The Moving Sofa Problem" video? Edit: That Reed-Solomon video finally came after 1 1/2 months.

[@fakhermokadem11]

2:42, Only Prof. Brailsford will make counting from 0 up so badass.

[@GordonjSmith1]

Very enlightening!

[@blahfasel2000]

The hyphens in the ISBN aren't arbitrary, for the old ISBN-10 system they separate the number into four blocks, the group number (1 to 5 digits, mostly a country code, but for some languages widely spoken in multiple countries there is a language code instead, for example 0 or 1 for English, and countries where those languages are primarily spoken generally don't have their own country code, so there isn't for example a code for the UK or the US), the publisher number (1 to 7 digits, assigned by regional organisations, publishers can have multiple codes if they exceed their initial assignment, or through acquisitions for example), the title number (1 to 7 digits, assigned by the publisher), and finally the check digit. The length of the different blocks of course always has to add up to 10, so you can't have for example a 3 digit group number and a 6 digit published number, as that would leave no room for the title number. For the new ISBN-13 the division is mostly the same, except that there is a new prefix added at the front which is (at the moment) either 978 or 979. The 978 prefix encompasses all the old ISBN-10s, you simply add the prefix in front of the old ISBN and replace the check digit with a EAN-13 check digit instead (this corresponds to the old "bookland" encoding of ISBNs into EAN-13 barcodes). The only group numbers with a 979 prefix currently allocated are "979-10" for French (the language, not France as a country!), "979-11" for South Korea (this is a country, not a language code), and "979-12" for Italy (however also used for italian language books published in Switzerland)

[@danielsharp2402]

I'll have an exam involving reed solomon codes in a week, perfect timing :)

[@SebastiaanDingemans]

1:17 international standard book numbers book codes

[@5ilver42]

*MORE BRAILSFORD!!!*

[@andygolem5514]

The issue I found with set theory is very fundamental i.e. 011 = 01 we can discard positional info for brevity but I thought it had to be useful to consider position as well i.e. that the placement of elements can also be mathematically useful, though I'm too lazy to figure out how.

[@quesoestbonne]

There's a similar check digits on UK VAT numbers. Sum of weighted digits for the first 9 then modulo 98 remainder forms the last two. (If I remembered correctly)

[@pierreabbat6157]

X is a bad choice to denote an erasure in ISBN, because X means 10 in the checksum.

[@profdaveb6384]

Hmmm! OK. But I did use *lower case" x to denote the erasure whereas in ISBN checksum if x occurs it's required (I think) that it be an upper-case X.

[@adammercer9679]

When he was talking about doing the basic arithmetic operations and mentioned that division wasn't in the set, was he referring to a set being closed under those operations? If so, based on the set he gave us, wouldn't it be impossible to be closed under subtraction? For instance, "n - (n + 1)" would give us a negative integer and that would take us outside the "field."

[@gordonrichardson2972]

In modular arithmetic the numbers 'wrap around', so the result of subtraction is always in the field.

[@colinstu]

15:43 what's that CD jewel case seen to the right of the lamp / left of the monitor?

[@colinstu]

20:58...answered my own question

[@0707andy]

Anyone: mentions 2^8 Computer scientist : TRIGGERED

[@RamkrishanYT]

Why?

[@felipecramos00]

@@RamkrishanYT 2^8 = 256 -> number of possible combinations that a byte can have (that's what i think)

[@lawrencedoliveiro9104]

I’ve wondered why telephone numbers don’t have check digits. It would reduce the incidence of wrong numbers by 90%. Yes, it would make them longer. That mattered more back when they might only have been 5-6 digits. Now that they are commonly 7-8 digits or longer (particularly for mobile phone users), it seems less of an issue. Come to think of it, why are we still using telephone numbers? They’re an idea that dates from the 19th century. Why don’t we use 20th century Internet-based technologies, such as the Domain Name System? Use names, instead of numbers, to connect to people.

[@tamasdemjen4242]

This. In addition, why do I have to spell my name, when I could just type it in. Again, 19th century technology.

[@lawrencedoliveiro9104]

@@tamasdemjen4242 We have technologies like autocomplete. It’s like speed-dial for the 21st century.

[@TrueThanny]

Let me know how that works out when you instruct your phone to call "John Smith". DNS host names are unique. Human names are not. Quite apart from that, we already have that functionality, and have had it for quite some time. It's called an address book, whether a literal book next to the phone, or a digital version in the phone's memory.

[@RussellTeapot]

@@TrueThanny ahahah you are right! I never thought about address books like DNS: in fact, they map names and addresses (in a unique fashion) to numbers

[@lawrencedoliveiro9104]

@@TrueThanny Yes, address books are precisely my point. Why do we have to do the lookup of the number, and then have to dial that number? Why not have the computer do the lookup for us? That’s how the DNS works -- it’s like an address book that automatically keeps itself up-to-date!

[@NikolajLepka]

In your example with positive integers including zero: You can't "reliably" subtract either. Just like how 4/2 = 2, i.e. a positive integer, you can of course also subtract and get a positive integer: 5 - 2 = 3 However, flip the numbers, and your result is suddenly no longer positive: 2 - 5 = -3. The numbers you _should_ have used were all the integers -- that is, including negatives -- thereby getting around the issue. Also, side note: integers always include zero. What you are referring to are called "natural numbers"

[@NikolajLepka]

@Johan Gustafsson so does division after you add modulo into the mix Your argument is invalid

[@NikolajLepka]

@Johan Gustafsson and what part of my argument is false? The video shows that both division and subtraction works in the realm of natural numbers after modulo is added in. My argument revolves entirely around integers without modulo.

[@PvblivsAelivs]

Well, you can get "multiplicative inverses" for what many would consider unusual definitions of addition and multiplication.

[@johnfrancisdoe1563]

John Undefined And GF[256] is a very useful such unusual definition.

[@PvblivsAelivs]

@@johnfrancisdoe1563 I am familiar with it (or rather them as there are several irreducible polynomials that produce different fields.) But calling the operations "addition" and "multiplication" goes very much against the grain. I, for one, would rather see other terms used.

[@johnfrancisdoe1563]

John Undefined It's completely normal in that branch of computer science and math. Just like "straight line" and "right angle" are commonly used in descriptions of non-Euclidian geometry.

[@DanBoulet]

It also works if you reverse the order of the weights, start at 10 in the first position and make your way down to 1.

[@johnfrancisdoe1563]

Dan Boulet Or go randomly.

[@rishi9881]

There is a mistake. At 7:39, it should be "remainder of 1", and not "remainder of 2" as stated by the professor.

[@profdaveb6384]

Thanks for spotting that one! Despite my best efforts these slip-ups occur from time to time. I'm currently working on the subtitles for this video and will insert [correction: "remainder of 1" ] at this 7:39 point.

[@adrianperez8695]

@@profdaveb6384 Just wanted to let you know, your sections on Computerphile helped me to realize that computer science is a profession I can pursue. Thank you for the passion you show in your videos.

[@XSpamDragonX]

Couldn't a book search tool see that the given ISBN is wrong, do error correction for each of the 10 digits individually, and offer a list of any books that match?

[@IceMetalPunk]

What if your ISBN has a 2 in position 6 and a 6 in position 2, but you accidentally transpose them? Would you not even be able to detect the error?

[@Nixitur]

No, you would, because a transposition of that would be a 2 in position 2 and a 6 in position 6. The relevant part of the sum of your example would be 2 * 6 + 6 * 2 = 1 + 1 = 2 but transposed, it would be 2 * 2 + 6 * 6 = 4 + 3 = 7 The sums are not the same, so you can detect the error.

[@GoatzAreEpic]

5:19 everything is absolutely honkiedory XDDD

[@electronash]

This video goes all the way up to 11.

[@Mr.Beauregarde]

I'm a little perturbed at the Equivocation existing in this video. The inverse and the identity, despite being approximately similar in this instance, are definitely not equivalent.

[@frognik79]

That division sequence looks like an xor.

[@johnfrancisdoe1563]

frognik79 Not quite. In GF[2ⁿ], XOR is actually the addition. However in GG[11ⁿ], addition is addition modulo 11 of each element. Though it may or may not be possible to use XOR gates in a hardware implementation of Modulo 11 reciprocal, if you use a tech where XOR gates are cheaper than multiple NAND or NOR gates. But it's been almost 4 decades since doing decimal check digits with a microprocessor became efficient and cheap enough not to bother with inflexible hardware (because the number of codes processed and amount of other per item processing would dwarf any technical advantage). Binary codes like CRC, RS etc. are used for much more gigantic data volumes often at the speed critical path of systems, so those are routinely done in hardware.

[@U014B]

Galois, promptly after getting shot: "My disappointment is immeasurable, and my day is ruined."

[@johnfrancisdoe1563]

Noel Goetowski Really? Someone else gave a different quote.

[@thudso]

To be that pedantic mathematician: the integers are not a field, they are a ring, and at the beginning of the video the professor uses the natural numbers which are called N (he should also include -1,-2,-3...etc to define Z). Everything else looks fine, just don't rely on the first few minutes for an exam!

[@AlcuBerry]

Wasn't this literally Numberphile's first video? 🙃

[@ThreeBlackCatsInaAlley]

When I'll be able to understand this?

[@VoilaTadaOfficial]

When you can quickly follow mod calculations.

[@YuKonSama]

if you only want to have non-negative integers, subtraction is a problem too, according to his logic. 5-3 is not a non-negativ number...

[@Elesario]

Naughty, you didn't blur out his number plate.

[@christophermcclellan8730]

Wait! No. The video can’t be over yet. You’ve not shown me how this works with powers of 2 or where I could read more! Google searches talk about the duel or the raw maths, not how it applies to error correction.

[@greatestuff]

Computerphile left me on a cliffhanger ...where's the reed solomon codes?

[@johnfrancisdoe1563]

Courtland Colburn At least 10 chapters and lectures later in the full course, as far as I recall.

[@EmbeddedSorcery]

This is so confusing... If each digit of the ISBN means something... like "language" and such, then how can we just manipulate them algebraically and get a meaning out of it? They have leeway in the "meaning" behind the code in order to choose mod 11 numbers?

[@johnfrancisdoe1563]

JK W The last digit has no meaning but to check you heard/typed the others right. Human language is similar, but not mathematically optimized, which is why we can often understand garbled and incomplete sentences.

[@bkboggy]

You lost me at 00:01.

[@RandomGeometryDashStuff]

17:18 what is multiplicative inverse of 4 mod 16?

[@jekyllgaming99]

The multiplicative inverse of 4 mod 16 is 13. Simplifying a lot for the sake of concision, but here we go: The numbers mod 16 are elements of GF(16). The elements of GF(16) can be represented in binary as (ax^3 + bx^2 + cx + d), where {a, b, c, d} are either 0 or 1, and x^4 = x + 1. Multiplying 4 (0100) by 13 (1101) we get 0100000 + 010000 + 0100 = 110100; For clarity, the sum is done using bitwise-XOR as GF(16) is created from GF(2). 110100 = x^5 + x^4 + x^2 = (x * x^4) + x^4 + x^2. Substituting x^4 = x + 1 from earlier, this becomes x(x + 1) + (x + 1) + x^2, or 0110 + 0011 + 0100 = 0001 (again, using bitwise-XOR). There's a fair bit I've not explained here (mostly as I've only learned about it in the last couple hours), but it is a fascinating subject to dive into.

[@RandomGeometryDashStuff]

​@@jekyllgaming99did you redefine multiplication and mod? with normal multiplication and mod (like in video): 4*13 = 4 + 3*16 (4*13) mod 16 = 4 4 ≠ 1

[@jekyllgaming99]

@@RandomGeometryDashStuff The issue is that the base you're using (16) is non-prime - using modular arithmatic on a base of 16, division cannot work as 4X mod 16 ≠ 1 for all X ∈ ℤ . However, since 16 is a power of 2 (a prime number), you can construct a Galois field in which division can work using modulo-2 arithmetic. This necessitates using a polynomial of order 4 (as 16 is 2^4) that is primitive to GF(2), so that every element can be generated from it. There are 2 options for a base of 2^4, but the one commonly chosen is X^4 + X + 1, generating the galois field GF(16) as the quotient ring GF(2)[X]/(X^4 + X + 1), the elements of which can be represented as ax^3 + bx^2 + cx + 1, where a,b,c,d are either 0 or 1 (elements of GF(2)) and x is a root of X^4 + X + 1 i.e. x^4 = x + 1. This states that 16 (0b10000, represented as x^4) is equivalent to 3 (0b11, represented as x + 1). There's a fair bit of detail that would be hard to explain in a youtube comment, but essentially 13 is the multiplicative inverse of 4 in base 16 due to the modulo-2 arithmetic and x^4 = x + 1 equivalence of GF(16). If you were to choose the other polynomial to create GF(16), X^4 + X^3 + 1, the multiplicative inverse would be different (in this case, the multiplicative inverse of 4 would be 6). As an extra, here are the pairs of multiplicative inverses for each field: GF(2)[X]/(X^4 + X + 1) = (1,1) (2,9) (3,14) (4,13) (5,11) (6,7) (8,15) (10,12) GF(2)[X]/(X^4 + X^3 + 1) = (1,1) (2,12) (3,8) (4,6) (5,15) (7,14) (9,13) (10,11) He does go into a little more detail in the video 'Reed Solomon Encoding', but it's still quite rudimentary and worth delving into yourself to get a full understanding.

[@KemiksPL]

4:55 Shouldn't it start at 0?

[@nikolas9105]

There is no integer from 0-10 that you can multiply by 0 to make the modulo 11 of that number equal to 1

[@KemiksPL]

​@@nikolas9105 I meant the clock metaphor of mod 11. If I'm not mistaken, numbers on it should range from 0 to 10. I.e. 11 mod 11 = 0

[@nikolas9105]

@@KemiksPL Oh I think you are right

[@C00Cker]

11 is the same thing as 0 in (mod 11) so technically it doesn't matter, but sure programmers would argue calling it 0 is better for obvoius reasons.

[@nikolas9105]

@@KemiksPL Actually with the clock metaphor it makes sense even though there is no digit for 11 in base 11 since clocks have 12 instead of 0

[@Hauketal]

Too bad all this about ISBN is going to be of historical value only. The ISBN were allocated to publishers in much too large blocks at the beginning, so they ran out of available numbers. Even then, smaller blocks would've run out soon after. Very similar to the problem with IPv4 addresses.

[@dannyhendron2903]

ISBNv6 when?

[@mytech6779]

IPv6 gets a lot of resistance because the particulars of its implementation, in addition to be more complex than necessary. [k.i.s.s] IPv6 may further some of the current political attempts around the world, by those wishing to maintain a grip on power, to erode the semi-anonymity that made the web [and internet generally] foster a new era of education, information, and forthright [honest, politically unburdened] communications. In most of the more honest court systems, dynamic IPv4 address allocation and NAT (both made necessary by the address shortage) are easy enough in concept for non IT educated judges and juries to understand that IP addresses are not useful as proper evidence of who may have been behind the keyboard, especially when the two are combined.

[@Gidaio]

7:16 Literally the scariest finger I've ever seen. It's just... weird.

[@ianprado1488]

Steven Girvin, PhD wants to know your location

[@ShawnJonesHellion]

Have any of you thought of using symbols/ geometric shapes instead of numbers for code. Have not scientists claim there's some sort of geometric pattern to how everything is built. If this is the case if it's not the shape itself it's how the numbers are used to build the shape? Visual is faster recognized by humans then any sort of number problem therefore it seems more natural

[@alexjohnward]

he did at the start of the video with cubes

[@johnfrancisdoe1563]

Shawn Jones What scientists used to do many many years ago, was to use the _word_ "geometric" to really mean "mathematically structured with rules and proofs, just like in that ancient Greek book about geometry".

[@danieljensen2626]

Integers are a ring, not a field, precisely because you aren't guaranteed division.

[@fllthdcrb]

Oh yeah he did say, "field of integers".

[@richardeadon6396]

He keeps saying you can get every remainder up to 10 by dividing 11 by different integers. How is it possible to get a remainder greater than half the dividend (in this case 5)? 11/11 = 1 11/10 = 1 remainder 1 11/9 = 1 remainder 2 11/8 = 1 remainder 3 11/7 = 1 remainder 4 11/6 = 1 remainder 5 11/5 = 2 remainder 1 11/4 = 2 remainder 3 11/3 = 3 remainder 2 11/2 = 5 remainder 1 11/1 = 11

[@johnfrancisdoe1563]

Richard Eadon It's not the remainder of dividing by 11 by 5. It's the remainder (from dividing by 11) of dividing 1 by 5. Or ((1 / 5) remainder modulo 11).

[@brendawilliams8062]

And 256

[@user-iu1xg6jv6e]

Finally I can mark it as done: ☑ Hear Professor Brailsford says niþþle.

[@johnfrancisdoe1563]

ɐɯɹɐʞ ɐıuɐɯ Pronounced Niththle?

[@lawrencedoliveiro9104]

@@johnfrancisdoe1563 Ðat’s what I þought.

[@DancingRain]

Ðis is a þorny spelling issue :P

[@aDifferentJT]

In the natural numbers you can’t subtract

[@robertlinke2666]

why? natural numbers are positive intergers excluding 0, right? i can subtract 4 by 2. so i get 2. 4 and 2 are both natural numbers. i do not get a negative value. so why wouldn't i be able to?

[@fllthdcrb]

I think what Jonathan Tanner means is that natural numbers are not *closed* under subtraction. I.e. you can't subtract *any two natural numbers* and be _guaranteed_ to get *a natural number.* And that's whether or not you include 0 in the definition.

[@robertlinke2666]

@@fllthdcrb okay yeah that makews sense. i can do 4-2 but not 2-4. not every number will work.

[@lawrencedoliveiro9104]

That’s like saying that in the reals you can’t divide.

[@aDifferentJT]

There is no function f : N^2 -> N that represents subtraction

[@dosmastrify]

4:30 clock math

[@Garjahn]

Hah, i mistook the thumbnail of this video for an internet historian video at a glance.

[@aDifferentJT]

Galois fields are used in QR codes

[@wumpusthehunted2628]

Jonathan Tanner and anything using Reed Solomon codes (all optical discs do, and it was state of the art through the 1990s).

[@Jirayu.Kaewprateep]

📺💬 You don't need to worry about the hyphen as long as the math is concerned. 🥺💬 Mathematics is to solve the problem within the pattern or formatting we cannot resolve for the problem we cannot define. 🐑💬 The series is from the results by divided by Prime number and how about two or more prime numbers in a series when 2 prim numbers summation are a prime numbers or the output remains the property. 🥺💬 The result from multiplication from two identity can see as number but remain a property ex. 61 x 63 x 20 = 39060 🐑💬 And you would know when you holding the 🔑 Key which is Prime number

[@alwysrite]

so the question is why do they use mod 11 at all? why not just a serial number? what does it achieve?

[@intriga.realmedia]

I want to be like him..

[@BenDover-xd1jh]

Null

[@Simbosan]

Presumably if two numbers were missing, you would be stuffed as there could be many possible solutions

[@carly09et]

You can recover three numbers - just :), the code has a context which gives ancillary keys to reduce the errors.

[@johnfrancisdoe1563]

Finian Blackett No. The number of correctable errors is strictly limited by the Hamming distance, which is strictly limited by the number of check digits added, limit is higher for erasure codes than for codes that must also detect the errors. RS codes can be systematically designed for almost any number of errors, adding more check digits to do so, but the fundamental limits remain. To fix N missing digits, you need at least N check digits before the loss. To detect and fix up to N errors, you need enough check digits to count all the possibilities, including the possibility of no error at all.

[@Simbosan]

@@johnfrancisdoe1563Thanks, i don't have the knowledge to reply to Finian's assertion, but obviously there is no strategy for detecting arbitrary numbers of errors. If there was the implications for maths and logic would be err... large?

[@kenichimori8533]

ISBN CODER 3 DIGIT.

[@brendawilliams8062]

As long as it’s 2666666

[@bestformspielt]

It goes to eleven!

[@PatFarrellKTM]

Little Feat! Very cool. Age, around US retirement age :-)

[@PhilBoswell]

Let it roll!

[@smjpl]

I'm around half that and came across dixie chicken recently. Cool song!

[@vk3fbab]

We can tell who watched all the way to the end. Little great a great band.

[@davesextraneousinformation9807]

Heads Hands and Feet, anyone?

[@koppadasao]

Oh, megabyte me!

[@bluekeybo]

Too much talking, too unnecessarily long. James Grime did it better

[@gordonrichardson2972]

James Grime (Numberphile) is a mathematician, Professor Brailsford is a computer scientist. Different audience.

[@johnfrancisdoe1563]

Gordon Richardson And Professor Brailsford pointed this out himself in the video.

[@profdaveb6384]

Yes ! Thank you.