Best Known (78−16, 78, s)-Nets in Base 2
(78−16, 78, 152)-Net over F2 — Constructive and digital
Digital (62, 78, 152)-net over F2, using
- 22 times duplication [i] based on digital (60, 76, 152)-net over F2, using
- trace code for nets [i] based on digital (3, 19, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- trace code for nets [i] based on digital (3, 19, 38)-net over F16, using
(78−16, 78, 268)-Net over F2 — Digital
Digital (62, 78, 268)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(278, 268, F2, 2, 16) (dual of [(268, 2), 458, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(278, 536, F2, 16) (dual of [536, 458, 17]-code), using
- 1 times truncation [i] based on linear OA(279, 537, F2, 17) (dual of [537, 458, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(273, 513, F2, 17) (dual of [513, 440, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 513 | 218−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(255, 513, F2, 13) (dual of [513, 458, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 513 | 218−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(26, 24, F2, 3) (dual of [24, 18, 4]-code or 24-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(279, 537, F2, 17) (dual of [537, 458, 18]-code), using
- OOA 2-folding [i] based on linear OA(278, 536, F2, 16) (dual of [536, 458, 17]-code), using
(78−16, 78, 3229)-Net in Base 2 — Upper bound on s
There is no (62, 78, 3230)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 302293 168793 646052 776737 > 278 [i]