Best Known (84−42, 84, s)-Nets in Base 32
(84−42, 84, 224)-Net over F32 — Constructive and digital
Digital (42, 84, 224)-net over F32, using
- 2 times m-reduction [i] based on digital (42, 86, 224)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (9, 31, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- digital (11, 55, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- digital (9, 31, 104)-net over F32, using
- (u, u+v)-construction [i] based on
(84−42, 84, 513)-Net in Base 32 — Constructive
(42, 84, 513)-net in base 32, using
- base change [i] based on digital (28, 70, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
(84−42, 84, 657)-Net over F32 — Digital
Digital (42, 84, 657)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3284, 657, F32, 42) (dual of [657, 573, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3284, 1038, F32, 42) (dual of [1038, 954, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(36) [i] based on
- linear OA(3280, 1024, F32, 42) (dual of [1024, 944, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(3270, 1024, F32, 37) (dual of [1024, 954, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(324, 14, F32, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(41) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(3284, 1038, F32, 42) (dual of [1038, 954, 43]-code), using
(84−42, 84, 293572)-Net in Base 32 — Upper bound on s
There is no (42, 84, 293573)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 2 707831 805721 371766 778802 959862 976474 861446 826133 373440 248674 347292 868656 964580 511483 481410 175319 685135 930891 272668 890666 373824 > 3284 [i]