A time span with stops (e.g. intervals between events) is isomorphic to "path" from graph theory. A path is a graph without branches; i.e. a graph with two nodes of degree one (first and last), and the rest of degree two.
Irreducible polynomial: a one that can't be reduced into multiple polynomials of a lower degree within the same field. Also, if a polynomial ƒ(x) have a root, that is ∃a, ƒ(a) = 0 then x-a is a factor of ƒ(x), i.e. ƒ(x) can be reduced.
A field consists of commutative addition and product operations, and the product operation distributes over an addition.
order— is a number of elements.galois fieldakafinite field— a field with finite number of elements. E.g. calculations GF(256) = GF(2⁸) are often used to convert bytes to bytes.hamming distance(d): minimal number of steps required to change one word to another, where "step" is a minimal replacement of a character. E.g. a subset = {1000, 0001} of a set with 4-bit words, has d = 4. On the other hand, a subset {0000, 0001} has d = 1. Letrbe a number of errors occured. Some facts:- Finding errors requires
d >= r - Correcting errors requires
d > 2r
- Finding errors requires
generator elementan element whose exponents generate every element of the group.cyclic groupa group withgenerator element
Finite fields typically made out of prime number of elements. Otherwise multiplication might have bad side-effects.
erasurean error in known position.hamming codes: are using "control codes" at every power of 2 position that controls parity of a group of the information. Not really effective.codewordis a data + redundancy.
Quoting: «It turns out that there is a finite field with p d elements for every prime number p and every positive integer d». Whatever that means, a paper says that commonly Reed-Solomon is based on GF(2^8) = GF(256). It says, the RS concept applies to any field.
linear code let a vector M = [x_1, .., x_m] be message vector, and a m×n matrix C with entries in the field; then MC = [x_1,…,x_n] is a linear code with m message symbols, and n codeword symbols. In an example[1] first m columns of C is an identity matrix m×m; and as results MC is M|P, i.e. the original M, and some redundancy info P.
Exponent in GF(x^n) signifies a number of elements side-by-side. E.g. GF(2^8) consist of 8 numbers one-by-one.
Addition & multiplication are not the usual ones. Multiplication works as (a×b) mod c, where × is a usual multiplication, mod is a division remainder, and c is an irreducible polynom of power one greater then number of bits.
Basically, works like following: a sequence of bits, e.g. 1111 being represented as a polynom 1 * x^3 + 1 * x^2 + 1 * x^1 + 1 * x^0. Then we find an irreducible polynom of a degree n higher, which we use to encode information. Then we can lose n bits of information, and, with some magic, still restore the 1111. The magic of decoding is a big research field, with a bunch of different algos, and, seemingly, none is easy to describe.
Embedding: an injection from smaller object to a bigger. In particular there could be many embeddings. See also "word embeddings".Hamming distance: given two equally-sized words, it's a num of substution operations required to change one into another.Levenshtein distance: given two words, it's a num of {insertion,deletion,substution} operations required to change one into another.
1: Notes on Reed-Solomon codes.
- Number of permutations of n is
n! - Number of k permutations of n is
n! / (n-k)!.
- when one probability leads to another, which leads to third, they are all multiplied
- when one probability leads to two different that both are of interest, the two are summed up