Best Known (33, 33+4, s)-Nets in Base 2
(33, 33+4, 262145)-Net over F2 — Constructive and digital
Digital (33, 37, 262145)-net over F2, using
(33, 33+4, 262163)-Net over F2 — Digital
Digital (33, 37, 262163)-net over F2, using
- net defined by OOA [i] based on linear OOA(237, 262163, F2, 4, 4) (dual of [(262163, 4), 1048615, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(237, 262163, F2, 3, 4) (dual of [(262163, 3), 786452, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(237, 262163, F2, 4) (dual of [262163, 262126, 5]-code), using
- 1 times truncation [i] based on linear OA(238, 262164, F2, 5) (dual of [262164, 262126, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(237, 262144, F2, 5) (dual of [262144, 262107, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(219, 262144, F2, 3) (dual of [262144, 262125, 4]-code or 262144-cap in PG(18,2)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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(238, 262164, F2, 5) (dual of [262164, 262126, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(237, 262163, F2, 4) (dual of [262163, 262126, 5]-code), using
- appending kth column [i] based on linear OOA(237, 262163, F2, 3, 4) (dual of [(262163, 3), 786452, 5]-NRT-code), using
(33, 33+4, 524285)-Net in Base 2 — Upper bound on s
There is no (33, 37, 524286)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 137439 215614 > 237 [i]