Best Known (78, 78+16, s)-Nets in Base 2
(78, 78+16, 260)-Net over F2 — Constructive and digital
Digital (78, 94, 260)-net over F2, using
- 2 times m-reduction [i] based on digital (78, 96, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 24, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 24, 65)-net over F16, using
(78, 78+16, 747)-Net over F2 — Digital
Digital (78, 94, 747)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(294, 747, F2, 2, 16) (dual of [(747, 2), 1400, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(294, 1038, F2, 2, 16) (dual of [(1038, 2), 1982, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(294, 2076, F2, 16) (dual of [2076, 1982, 17]-code), using
- 1 times truncation [i] based on linear OA(295, 2077, F2, 17) (dual of [2077, 1982, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(289, 2049, F2, 17) (dual of [2049, 1960, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(267, 2049, F2, 13) (dual of [2049, 1982, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- 1 times truncation [i] based on linear OA(295, 2077, F2, 17) (dual of [2077, 1982, 18]-code), using
- OOA 2-folding [i] based on linear OA(294, 2076, F2, 16) (dual of [2076, 1982, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(294, 1038, F2, 2, 16) (dual of [(1038, 2), 1982, 17]-NRT-code), using
(78, 78+16, 12954)-Net in Base 2 — Upper bound on s
There is no (78, 94, 12955)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 19818 025398 496909 054433 123507 > 294 [i]