Best Known (37, 37+4, s)-Nets in Base 2
(37, 37+4, 1048577)-Net over F2 — Constructive and digital
Digital (37, 41, 1048577)-net over F2, using
(37, 37+4, 1048597)-Net over F2 — Digital
Digital (37, 41, 1048597)-net over F2, using
- net defined by OOA [i] based on linear OOA(241, 1048597, F2, 4, 4) (dual of [(1048597, 4), 4194347, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(241, 1048597, F2, 3, 4) (dual of [(1048597, 3), 3145750, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(241, 1048597, F2, 4) (dual of [1048597, 1048556, 5]-code), using
- 1 times truncation [i] based on linear OA(242, 1048598, F2, 5) (dual of [1048598, 1048556, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(241, 1048576, F2, 5) (dual of [1048576, 1048535, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(221, 1048576, F2, 3) (dual of [1048576, 1048555, 4]-code or 1048576-cap in PG(20,2)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(221, 22, F2, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,2)), using
- dual of repetition code with length 22 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- 1 times truncation [i] based on linear OA(242, 1048598, F2, 5) (dual of [1048598, 1048556, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(241, 1048597, F2, 4) (dual of [1048597, 1048556, 5]-code), using
- appending kth column [i] based on linear OOA(241, 1048597, F2, 3, 4) (dual of [(1048597, 3), 3145750, 5]-NRT-code), using
(37, 37+4, 2097149)-Net in Base 2 — Upper bound on s
There is no (37, 41, 2097150)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2 199024 304126 > 241 [i]