Best Known (163−31, 163, s)-Nets in Base 2
(163−31, 163, 263)-Net over F2 — Constructive and digital
Digital (132, 163, 263)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (0, 15, 3)-net over F2, using
- net from sequence [i] based on digital (0, 2)-sequence over F2, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 0 and N(F) ≥ 3, using
- the rational function field F2(x) [i]
- Niederreiter sequence [i]
- Sobol sequence [i]
- net from sequence [i] based on digital (0, 2)-sequence over F2, using
- digital (117, 148, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 37, 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, 37, 65)-net over F16, using
- digital (0, 15, 3)-net over F2, using
(163−31, 163, 537)-Net over F2 — Digital
Digital (132, 163, 537)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2163, 537, F2, 2, 31) (dual of [(537, 2), 911, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2163, 1074, F2, 31) (dual of [1074, 911, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2163, 1075, F2, 31) (dual of [1075, 912, 32]-code), using
- construction XX applied to C1 = C([1019,22]), C2 = C([0,26]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([1019,26]) [i] based on
- linear OA(2131, 1023, F2, 27) (dual of [1023, 892, 28]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,22}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2131, 1023, F2, 27) (dual of [1023, 892, 28]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2151, 1023, F2, 31) (dual of [1023, 872, 32]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,26}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2111, 1023, F2, 23) (dual of [1023, 912, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(26, 26, F2, 3) (dual of [26, 20, 4]-code or 26-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
- linear OA(26, 26, F2, 3) (dual of [26, 20, 4]-code or 26-cap in PG(5,2)) (see above)
- construction XX applied to C1 = C([1019,22]), C2 = C([0,26]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([1019,26]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2163, 1075, F2, 31) (dual of [1075, 912, 32]-code), using
- OOA 2-folding [i] based on linear OA(2163, 1074, F2, 31) (dual of [1074, 911, 32]-code), using
(163−31, 163, 11430)-Net in Base 2 — Upper bound on s
There is no (132, 163, 11431)-net in base 2, because
- 1 times m-reduction [i] would yield (132, 162, 11431)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 5 851651 030340 486442 572822 650361 810988 878933 264280 > 2162 [i]