Best Known (32−17, 32, s)-Nets in Base 32
(32−17, 32, 131)-Net over F32 — Constructive and digital
Digital (15, 32, 131)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- digital (7, 24, 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 (0, 8, 33)-net over F32, using
(32−17, 32, 260)-Net in Base 32 — Constructive
(15, 32, 260)-net in base 32, using
- base change [i] based on digital (3, 20, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
(32−17, 32, 261)-Net over F32 — Digital
Digital (15, 32, 261)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3232, 261, F32, 17) (dual of [261, 229, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3232, 341, F32, 17) (dual of [341, 309, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 341 | 322−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(3232, 341, F32, 17) (dual of [341, 309, 18]-code), using
(32−17, 32, 321)-Net in Base 32
(15, 32, 321)-net in base 32, using
- t-expansion [i] based on (14, 32, 321)-net in base 32, using
- base change [i] based on digital (2, 20, 321)-net over F256, using
- net from sequence [i] based on digital (2, 320)-sequence over F256, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 2 and N(F) ≥ 321, using
- net from sequence [i] based on digital (2, 320)-sequence over F256, using
- base change [i] based on digital (2, 20, 321)-net over F256, using
(32−17, 32, 82559)-Net in Base 32 — Upper bound on s
There is no (15, 32, 82560)-net in base 32, because
- 1 times m-reduction [i] would yield (15, 31, 82560)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 45676 061176 804675 396653 550902 190643 773918 870033 > 3231 [i]