Best Known (75−26, 75, s)-Nets in Base 16
(75−26, 75, 583)-Net over F16 — Constructive and digital
Digital (49, 75, 583)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 19, 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
- digital (30, 56, 518)-net over F16, using
- trace code for nets [i] based on digital (2, 28, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- trace code for nets [i] based on digital (2, 28, 259)-net over F256, using
- digital (6, 19, 65)-net over F16, using
(75−26, 75, 596)-Net in Base 16 — Constructive
(49, 75, 596)-net in base 16, using
- (u, u+v)-construction [i] based on
- (8, 21, 80)-net in base 16, using
- base change [i] based on digital (1, 14, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- base change [i] based on digital (1, 14, 80)-net over F64, using
- digital (28, 54, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 27, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 27, 258)-net over F256, using
- (8, 21, 80)-net in base 16, using
(75−26, 75, 3360)-Net over F16 — Digital
Digital (49, 75, 3360)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1675, 3360, F16, 26) (dual of [3360, 3285, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(1675, 4107, F16, 26) (dual of [4107, 4032, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- linear OA(1673, 4096, F16, 26) (dual of [4096, 4023, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(1664, 4096, F16, 23) (dual of [4096, 4032, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(162, 11, F16, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,16)), using
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- Reed–Solomon code RS(14,16) [i]
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(1675, 4107, F16, 26) (dual of [4107, 4032, 27]-code), using
(75−26, 75, 3343182)-Net in Base 16 — Upper bound on s
There is no (49, 75, 3343183)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 2 037040 175255 250070 758761 545685 758595 723810 449802 215246 380380 720618 042365 163688 245613 059986 > 1675 [i]