Best Known (126, 126+29, s)-Nets in Base 2
(126, 126+29, 265)-Net over F2 — Constructive and digital
Digital (126, 155, 265)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (1, 15, 5)-net over F2, using
- net from sequence [i] based on digital (1, 4)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 1 and N(F) ≥ 5, using
- Niederreiter–Xing sequence (Piršić implementation) with equidistant coordinate [i]
- net from sequence [i] based on digital (1, 4)-sequence over F2, using
- digital (111, 140, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 35, 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, 35, 65)-net over F16, using
- digital (1, 15, 5)-net over F2, using
(126, 126+29, 653)-Net over F2 — Digital
Digital (126, 155, 653)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2155, 653, F2, 3, 29) (dual of [(653, 3), 1804, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2155, 683, F2, 3, 29) (dual of [(683, 3), 1894, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2155, 2049, F2, 29) (dual of [2049, 1894, 30]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(2155, 2049, F2, 29) (dual of [2049, 1894, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2155, 683, F2, 3, 29) (dual of [(683, 3), 1894, 30]-NRT-code), using
(126, 126+29, 12361)-Net in Base 2 — Upper bound on s
There is no (126, 155, 12362)-net in base 2, because
- 1 times m-reduction [i] would yield (126, 154, 12362)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 22850 769456 448853 318784 347014 888750 922493 137184 > 2154 [i]