Best Known (45, 45+40, s)-Nets in Base 32
(45, 45+40, 240)-Net over F32 — Constructive and digital
Digital (45, 85, 240)-net over F32, using
- 6 times m-reduction [i] based on digital (45, 91, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 34, 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 (11, 57, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 34, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(45, 45+40, 513)-Net in Base 32 — Constructive
(45, 85, 513)-net in base 32, using
- 17 times m-reduction [i] based on (45, 102, 513)-net in base 32, using
- base change [i] based on digital (28, 85, 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
- base change [i] based on digital (28, 85, 513)-net over F64, using
(45, 45+40, 1012)-Net over F32 — Digital
Digital (45, 85, 1012)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3285, 1012, F32, 40) (dual of [1012, 927, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3285, 1050, F32, 40) (dual of [1050, 965, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(29) [i] based on
- linear OA(3276, 1024, F32, 40) (dual of [1024, 948, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3259, 1024, F32, 30) (dual of [1024, 965, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(329, 26, F32, 9) (dual of [26, 17, 10]-code or 26-arc in PG(8,32)), using
- discarding factors / shortening the dual code based on linear OA(329, 32, F32, 9) (dual of [32, 23, 10]-code or 32-arc in PG(8,32)), using
- Reed–Solomon code RS(23,32) [i]
- discarding factors / shortening the dual code based on linear OA(329, 32, F32, 9) (dual of [32, 23, 10]-code or 32-arc in PG(8,32)), using
- construction X applied to Ce(39) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(3285, 1050, F32, 40) (dual of [1050, 965, 41]-code), using
(45, 45+40, 668075)-Net in Base 32 — Upper bound on s
There is no (45, 85, 668076)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 86 648004 910101 301176 698156 838855 150477 895707 459668 403369 597310 664338 459203 114659 105521 516384 890962 292477 218987 147626 283408 609756 > 3285 [i]