Best Known (37−17, 37, s)-Nets in Base 32
(37−17, 37, 175)-Net over F32 — Constructive and digital
Digital (20, 37, 175)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (5, 13, 77)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (1, 9, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- digital (0, 4, 33)-net over F32, using
- (u, u+v)-construction [i] based on
- 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 (5, 13, 77)-net over F32, using
(37−17, 37, 301)-Net in Base 32 — Constructive
(20, 37, 301)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- (11, 28, 257)-net in base 32, using
- base change [i] based on (3, 20, 257)-net in base 128, using
- 4 times m-reduction [i] based on (3, 24, 257)-net in base 128, using
- base change [i] based on digital (0, 21, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 21, 257)-net over F256, using
- 4 times m-reduction [i] based on (3, 24, 257)-net in base 128, using
- base change [i] based on (3, 20, 257)-net in base 128, using
- digital (1, 9, 44)-net over F32, using
(37−17, 37, 842)-Net over F32 — Digital
Digital (20, 37, 842)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3237, 842, F32, 17) (dual of [842, 805, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3237, 1038, F32, 17) (dual of [1038, 1001, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(3233, 1024, F32, 17) (dual of [1024, 991, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(3223, 1024, F32, 12) (dual of [1024, 1001, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(16) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(3237, 1038, F32, 17) (dual of [1038, 1001, 18]-code), using
(37−17, 37, 720279)-Net in Base 32 — Upper bound on s
There is no (20, 37, 720280)-net in base 32, because
- 1 times m-reduction [i] would yield (20, 36, 720280)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1 532505 358814 264025 490574 743562 413038 232966 646585 004922 > 3236 [i]