poly-code
poly-code is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
At its defaults
show: "erase"
show: "distance"
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- the symbol lost at position 1 came back the same ×7
- the alphabet is a prime between 5 and 17 ×2
- the number of message symbols is a whole number between 2 and 4 ×2
- the number of symbols transmitted is a whole number between 3 and 12 ×2
- any k of the n symbols recover the message exactly ×1
- at least k symbols survive, or nothing can be recovered and the figure would be a lie ×1
- every choice of surviving symbols was tried ×1
- every lost position is one of the positions sent ×1
- every non-zero remainder has a reciprocal mod a prime ×1
- more symbols are sent than the message has ×1
- no position is lost twice ×1
- only the zero message gives the zero codeword ×1
- the code is shorter than the field and longer than the message ×1
- the code meets the Singleton bound exactly ×1
- the field is a prime between 3 and 31 ×1
- the lightest non-zero codeword weighs n − k + 1 ×1
- the message is k symbols of the alphabet ×1
- the survivors recover the message ×1
- there are at most as many evaluation points as field elements ×1
- with one symbol too few, exactly p messages fit the survivors ×1
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A polynomial through the gaps
Write the message as the coefficients of a polynomial and send its values instead. Any k of them determine the polynomial, so it does not matter which ones are lost — and it does not matter how many, as long as k survive.
ComputationEvery element is a power of one of them
Pick the right element of a finite field and its powers run through every other non-zero element exactly once before returning to one. Multiplication becomes addition of exponents, and a table of q − 1 entries replaces the whole multiplication table.
ComputationThe field with four elements
The integers modulo four are not a field: two times two is zero and two has no reciprocal. There is nevertheless a field with four elements, and building it means giving up on counting as the way to make arithmetic finite.