For anyone wondering how this relates to other parts of math: the integers and univariate polynomials like shown here are both rings under addition and multiplication (they'd just be monoids if only referring to one of the operations). In the modular arithmetic notation, the divisor on the right of the slash is known as an ideal (which is the equivalent of a normal subgroup from group theory). In the context of a homomorphism like `Z / 2Z`, the `2Z` ring is the kernel of the homomorphism (group/ring preserving transformation) because it's subset in the domain that gets mapped to the identity element in the codomain (zero for modular arithmetic). What's left over from the homomorphism in the codomain set after "removing" the equivalence class of cosets typified by the kernel ideal is homomorphism's image `Z / 2Z` or Z_2. You can see why prime-number-order integers under modular addition & multiplication are fields (being groups, not just monoids, as respects each operation) because the identities for each operation, 0 and 1 respectively, are their own inverses but for the rest of the numbers, they have to be able to be paired up so there must be an even number. Other than 2, all prime numbers are one more than an even number so you can expect that a field under modular arithmetic will necessarily be one more than an even order. But not all odd ordered sets of integers are fields so it's not sufficient. Maybe it has to do with composite numbers having non-trivial factors and the ability to break groups into simple group factors. Anyways, a year ago I had no idea what any of these words meant. So it's satisfying to see the puzzle coming together!
Love your videos. Here since tensors. One constructive criticism: please consider changing wording at the end. Instead of "supporting me" (that does not sound right) say something like "supporting my work" or "supporting this project" or "supporting this and future projects" or some such. I mean, basically, this is not about you, but the (great) work you're doing.
you just saved my computer science degree with your videos on block codes
You do an amazing job deriving complex topics from the ground up. I'm so glad to have access to this precious content!
You seriously need more subscribers! This information is too good!
For anyone wondering how this relates to other parts of math: the integers and univariate polynomials like shown here are both rings under addition and multiplication (they'd just be monoids if only referring to one of the operations). In the modular arithmetic notation, the divisor on the right of the slash is known as an ideal (which is the equivalent of a normal subgroup from group theory). In the context of a homomorphism like `Z / 2Z`, the `2Z` ring is the kernel of the homomorphism (group/ring preserving transformation) because it's subset in the domain that gets mapped to the identity element in the codomain (zero for modular arithmetic). What's left over from the homomorphism in the codomain set after "removing" the equivalence class of cosets typified by the kernel ideal is homomorphism's image `Z / 2Z` or Z_2.
You can see why prime-number-order integers under modular addition & multiplication are fields (being groups, not just monoids, as respects each operation) because the identities for each operation, 0 and 1 respectively, are their own inverses but for the rest of the numbers, they have to be able to be paired up so there must be an even number. Other than 2, all prime numbers are one more than an even number so you can expect that a field under modular arithmetic will necessarily be one more than an even order. But not all odd ordered sets of integers are fields so it's not sufficient. Maybe it has to do with composite numbers having non-trivial factors and the ability to break groups into simple group factors.
Anyways, a year ago I had no idea what any of these words meant. So it's satisfying to see the puzzle coming together!
can someone tell me 3:00 what value will be plugged into x in this case ? the order of the galois field ? the generator polynomial ? something else 😵
IIUC Being able to factor some polynomials only in complex coefficients is similar to how primes can be factored in rational/real/complex parts.
This is a great explanation. Great work!
at time 12:40, shouldn't the remainder be -x+1?
as we work in binary space -x = x :)
High-quality work. Thank you so much
Love your videos. Here since tensors. One constructive criticism: please consider changing wording at the end. Instead of "supporting me" (that does not sound right) say something like "supporting my work" or "supporting this project" or "supporting this and future projects" or some such. I mean, basically, this is not about you, but the (great) work you're doing.
doesn't supporting him mean that you are supporting his work?
Tres bon video! T'as m'aide bcp, merci
best video on this topic!!!
Ben Eater is also doing a series about error correcting algorithms and has made a video on CRC. Highly recommend it.
thank you for these very good videos
Where are remaining ECC videos?
I lost interested and never finished them. Sorry.
FREAKING LOVE YOU
Thank you it's really helpful
Ty very much'! You've helped so much
hello eigen Chris.. please upload further about tensors.. To go for understanding of General relativity
thanks
HANDS DOWN IM GIVING you a million bucks when I get rich
i spent a fucking hour in my lecture and got lost. You are god
Redundancy Space :-P
Wow!!!!!!!!!
it's different from other slide ,i have studied, use new ways to explain cyclic code,tks!!!