Best Known (45, 45+24, s)-Nets in Base 16
(45, 45+24, 581)-Net over F16 — Constructive and digital
Digital (45, 69, 581)-net over F16, using
- 1 times m-reduction [i] based on digital (45, 70, 581)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 18, 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 (27, 52, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 26, 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, 26, 258)-net over F256, using
- digital (6, 18, 65)-net over F16, using
- (u, u+v)-construction [i] based on
(45, 45+24, 643)-Net in Base 16 — Constructive
(45, 69, 643)-net in base 16, using
- (u, u+v)-construction [i] based on
- (9, 21, 129)-net in base 16, using
- base change [i] based on digital (0, 12, 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, 12, 129)-net over F128, using
- digital (24, 48, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
- (9, 21, 129)-net in base 16, using
(45, 45+24, 3170)-Net over F16 — Digital
Digital (45, 69, 3170)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1669, 3170, F16, 24) (dual of [3170, 3101, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(1669, 4107, F16, 24) (dual of [4107, 4038, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(1667, 4096, F16, 24) (dual of [4096, 4029, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(1658, 4096, F16, 21) (dual of [4096, 4038, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(23) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(1669, 4107, F16, 24) (dual of [4107, 4038, 25]-code), using
(45, 45+24, 2957734)-Net in Base 16 — Upper bound on s
There is no (45, 69, 2957735)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 121417 244230 523690 617023 247789 026182 089651 110014 643835 924008 043545 855670 459378 179676 > 1669 [i]