Best Known (100−35, 100, s)-Nets in Base 16
(100−35, 100, 585)-Net over F16 — Constructive and digital
Digital (65, 100, 585)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (13, 30, 71)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- digital (3, 20, 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
- digital (2, 10, 33)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (35, 70, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 35, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 35, 257)-net over F256, using
- digital (13, 30, 71)-net over F16, using
(100−35, 100, 643)-Net in Base 16 — Constructive
(65, 100, 643)-net in base 16, using
- (u, u+v)-construction [i] based on
- (13, 30, 129)-net in base 16, using
- base change [i] based on (3, 20, 129)-net in base 64, using
- 1 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- base change [i] based on digital (0, 18, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 18, 129)-net over F128, using
- 1 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- base change [i] based on (3, 20, 129)-net in base 64, using
- digital (35, 70, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 35, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 35, 257)-net over F256, using
- (13, 30, 129)-net in base 16, using
(100−35, 100, 3577)-Net over F16 — Digital
Digital (65, 100, 3577)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16100, 3577, F16, 35) (dual of [3577, 3477, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(16100, 4108, F16, 35) (dual of [4108, 4008, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(1697, 4096, F16, 35) (dual of [4096, 3999, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(1688, 4096, F16, 31) (dual of [4096, 4008, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(163, 12, F16, 3) (dual of [12, 9, 4]-code or 12-arc in PG(2,16) or 12-cap in PG(2,16)), using
- discarding factors / shortening the dual code based on linear OA(163, 16, F16, 3) (dual of [16, 13, 4]-code or 16-arc in PG(2,16) or 16-cap in PG(2,16)), using
- Reed–Solomon code RS(13,16) [i]
- discarding factors / shortening the dual code based on linear OA(163, 16, F16, 3) (dual of [16, 13, 4]-code or 16-arc in PG(2,16) or 16-cap in PG(2,16)), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(16100, 4108, F16, 35) (dual of [4108, 4008, 36]-code), using
(100−35, 100, 4921316)-Net in Base 16 — Upper bound on s
There is no (65, 100, 4921317)-net in base 16, because
- 1 times m-reduction [i] would yield (65, 99, 4921317)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 161391 024312 365541 011881 700759 068828 470130 128490 591302 430226 707039 856563 847789 370956 559439 602360 763528 090501 337407 110136 > 1699 [i]