Best Known (67−31, 67, s)-Nets in Base 32
(67−31, 67, 224)-Net over F32 — Constructive and digital
Digital (36, 67, 224)-net over F32, using
- 1 times m-reduction [i] based on digital (36, 68, 224)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (9, 25, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- digital (11, 43, 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 (9, 25, 104)-net over F32, using
- (u, u+v)-construction [i] based on
(67−31, 67, 306)-Net in Base 32 — Constructive
(36, 67, 306)-net in base 32, using
- (u, u+v)-construction [i] based on
- (6, 21, 129)-net in base 32, using
- base change [i] based on digital (0, 15, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 15, 129)-net over F128, using
- (15, 46, 177)-net in base 32, using
- 2 times m-reduction [i] based on (15, 48, 177)-net in base 32, using
- base change [i] based on digital (7, 40, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- base change [i] based on digital (7, 40, 177)-net over F64, using
- 2 times m-reduction [i] based on (15, 48, 177)-net in base 32, using
- (6, 21, 129)-net in base 32, using
(67−31, 67, 989)-Net over F32 — Digital
Digital (36, 67, 989)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3267, 989, F32, 31) (dual of [989, 922, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3267, 1044, F32, 31) (dual of [1044, 977, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(23) [i] based on
- linear OA(3261, 1024, F32, 31) (dual of [1024, 963, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3247, 1024, F32, 24) (dual of [1024, 977, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(30) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(3267, 1044, F32, 31) (dual of [1044, 977, 32]-code), using
(67−31, 67, 869082)-Net in Base 32 — Upper bound on s
There is no (36, 67, 869083)-net in base 32, because
- 1 times m-reduction [i] would yield (36, 66, 869083)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2187 261535 928258 292054 676253 592362 177576 938456 680978 060354 982332 912198 008990 228480 365065 112017 474800 > 3266 [i]