Best Known (45−22, 45, s)-Nets in Base 32
(45−22, 45, 174)-Net over F32 — Constructive and digital
Digital (23, 45, 174)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (5, 16, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- digital (7, 29, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (5, 16, 76)-net over F32, using
(45−22, 45, 288)-Net in Base 32 — Constructive
(23, 45, 288)-net in base 32, using
- 4 times m-reduction [i] based on (23, 49, 288)-net in base 32, using
- base change [i] based on digital (9, 35, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 35, 288)-net over F128, using
(45−22, 45, 540)-Net over F32 — Digital
Digital (23, 45, 540)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3245, 540, F32, 22) (dual of [540, 495, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3245, 1032, F32, 22) (dual of [1032, 987, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(3243, 1024, F32, 22) (dual of [1024, 981, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3237, 1024, F32, 19) (dual of [1024, 987, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(322, 8, F32, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3245, 1032, F32, 22) (dual of [1032, 987, 23]-code), using
(45−22, 45, 227548)-Net in Base 32 — Upper bound on s
There is no (23, 45, 227549)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 53 921203 563966 599857 107123 316400 528802 612863 583006 826719 435284 156060 > 3245 [i]