Best Known (90−31, 90, s)-Nets in Base 16
(90−31, 90, 596)-Net over F16 — Constructive and digital
Digital (59, 90, 596)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (13, 28, 82)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- digital (6, 21, 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 (0, 7, 17)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (31, 62, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 31, 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, 31, 257)-net over F256, using
- digital (13, 28, 82)-net over F16, using
(90−31, 90, 664)-Net in Base 16 — Constructive
(59, 90, 664)-net in base 16, using
- (u, u+v)-construction [i] based on
- (13, 28, 150)-net in base 16, using
- base change [i] based on digital (1, 16, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- base change [i] based on digital (1, 16, 150)-net over F128, using
- digital (31, 62, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 31, 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, 31, 257)-net over F256, using
- (13, 28, 150)-net in base 16, using
(90−31, 90, 3843)-Net over F16 — Digital
Digital (59, 90, 3843)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1690, 3843, F16, 31) (dual of [3843, 3753, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(1690, 4107, F16, 31) (dual of [4107, 4017, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- 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(1679, 4096, F16, 28) (dual of [4096, 4017, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(30) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(1690, 4107, F16, 31) (dual of [4107, 4017, 32]-code), using
(90−31, 90, 5972003)-Net in Base 16 — Upper bound on s
There is no (59, 90, 5972004)-net in base 16, because
- 1 times m-reduction [i] would yield (59, 89, 5972004)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 146784 213591 428213 333535 790048 003368 770898 742636 429804 526349 299215 890237 984377 595412 875158 542894 512059 552776 > 1689 [i]