Best Known (90−44, 90, s)-Nets in Base 32
(90−44, 90, 240)-Net over F32 — Constructive and digital
Digital (46, 90, 240)-net over F32, using
- 4 times m-reduction [i] based on digital (46, 94, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 35, 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, 59, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 35, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(90−44, 90, 513)-Net in Base 32 — Constructive
(46, 90, 513)-net in base 32, using
- 18 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
- base change [i] based on digital (28, 90, 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, 90, 513)-net over F64, using
(90−44, 90, 805)-Net over F32 — Digital
Digital (46, 90, 805)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3290, 805, F32, 44) (dual of [805, 715, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3290, 1044, F32, 44) (dual of [1044, 954, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(36) [i] based on
- linear OA(3284, 1024, F32, 44) (dual of [1024, 940, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [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(326, 20, F32, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,32)), using
- discarding factors / shortening the dual code based on linear OA(326, 32, F32, 6) (dual of [32, 26, 7]-code or 32-arc in PG(5,32)), using
- Reed–Solomon code RS(26,32) [i]
- discarding factors / shortening the dual code based on linear OA(326, 32, F32, 6) (dual of [32, 26, 7]-code or 32-arc in PG(5,32)), using
- construction X applied to Ce(43) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(3290, 1044, F32, 44) (dual of [1044, 954, 45]-code), using
(90−44, 90, 419673)-Net in Base 32 — Upper bound on s
There is no (46, 90, 419674)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 2907 443508 449008 874898 749594 698442 031013 047834 733997 936283 170525 680197 139888 839734 455485 994719 259069 141912 196345 061084 986535 179047 831404 > 3290 [i]