challenges / binary-codes-a-n-d
Largest binary codes A(n,d), open entries of Brouwer's table
Return an explicit binary code (list of integers below 2^n) of length n with minimum Hamming distance d, as large as you can, for 20 pairs (n, d) with n <= 28 whose best-known lower and upper bounds on A(n,d) still differ. The eval recomputes the minimum distance from the words; the score is code size over the best-known lower bound.
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# Largest binary codes A(n,d), open entries of Brouwer's table
## Goal
`code.py` exposes `code(n: int, d: int, time_budget: float, seed: int) -> list[int]`: a binary code
of length `n` and minimum Hamming distance `d`, given as a list of distinct integers in `[0, 2^n)`
(bit `i` of the integer is coordinate `i`). Return as many codewords as you can.
`A(n,d)` is the largest size of such a code. It is known exactly for small parameters and for a few
families (Hamming, Golay, Nordstrom-Robinson), but for most `(n, d)` with `n >= 17` the best known
code and the best proven upper bound are far apart. The 20 instances here are exactly such open
cells of Andries Brouwer's table of bounds on `A(n,d)` (the updated Best-Brouwer-MacWilliams-Odlyzko-
Sloane table), restricted to `n <= 28` and to sizes that verify quickly. A code larger than the
record in any of them is a new lower bound; the table has moved as recently as 2019 (Milshtein's
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