Best Known (31−12, 31, s)-Nets in Base 32
(31−12, 31, 264)-Net over F32 — Constructive and digital
Digital (19, 31, 264)-net over F32, using
- 1 times m-reduction [i] based on digital (19, 32, 264)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 33)-net over F32, using
- s-reduction based on digital (0, 1, s)-net over F32 with arbitrarily large s, using
- digital (0, 1, 33)-net over F32 (see above)
- digital (0, 2, 33)-net over F32, using
- digital (0, 2, 33)-net over F32 (see above)
- digital (0, 3, 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 (0, 4, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 6, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 13, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 1, 33)-net over F32, using
- generalized (u, u+v)-construction [i] based on
(31−12, 31, 684)-Net in Base 32 — Constructive
(19, 31, 684)-net in base 32, using
- 321 times duplication [i] based on (18, 30, 684)-net in base 32, using
- base change [i] based on digital (13, 25, 684)-net over F64, using
- net defined by OOA [i] based on linear OOA(6425, 684, F64, 12, 12) (dual of [(684, 12), 8183, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6425, 4104, F64, 12) (dual of [4104, 4079, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(6417, 4096, F64, 9) (dual of [4096, 4079, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- OA 6-folding and stacking [i] based on linear OA(6425, 4104, F64, 12) (dual of [4104, 4079, 13]-code), using
- net defined by OOA [i] based on linear OOA(6425, 684, F64, 12, 12) (dual of [(684, 12), 8183, 13]-NRT-code), using
- base change [i] based on digital (13, 25, 684)-net over F64, using
(31−12, 31, 2769)-Net over F32 — Digital
Digital (19, 31, 2769)-net over F32, using
(31−12, 31, 5773888)-Net in Base 32 — Upper bound on s
There is no (19, 31, 5773889)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 45671 940626 851276 678304 909713 758709 977711 218080 > 3231 [i]