Best Known (81, 81+24, s)-Nets in Base 32
(81, 81+24, 87414)-Net over F32 — Constructive and digital
Digital (81, 105, 87414)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 12, 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 (69, 93, 87381)-net over F32, using
- net defined by OOA [i] based on linear OOA(3293, 87381, F32, 24, 24) (dual of [(87381, 24), 2097051, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3293, 1048572, F32, 24) (dual of [1048572, 1048479, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3293, 1048576, F32, 24) (dual of [1048576, 1048483, 25]-code), using
- an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(3293, 1048576, F32, 24) (dual of [1048576, 1048483, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3293, 1048572, F32, 24) (dual of [1048572, 1048479, 25]-code), using
- net defined by OOA [i] based on linear OOA(3293, 87381, F32, 24, 24) (dual of [(87381, 24), 2097051, 25]-NRT-code), using
- digital (0, 12, 33)-net over F32, using
(81, 81+24, 174764)-Net in Base 32 — Constructive
(81, 105, 174764)-net in base 32, using
- base change [i] based on digital (51, 75, 174764)-net over F128, using
- 1281 times duplication [i] based on digital (50, 74, 174764)-net over F128, using
- net defined by OOA [i] based on linear OOA(12874, 174764, F128, 24, 24) (dual of [(174764, 24), 4194262, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(12874, 2097168, F128, 24) (dual of [2097168, 2097094, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(12874, 2097171, F128, 24) (dual of [2097171, 2097097, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(12874, 2097171, F128, 24) (dual of [2097171, 2097097, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(12874, 2097168, F128, 24) (dual of [2097168, 2097094, 25]-code), using
- net defined by OOA [i] based on linear OOA(12874, 174764, F128, 24, 24) (dual of [(174764, 24), 4194262, 25]-NRT-code), using
- 1281 times duplication [i] based on digital (50, 74, 174764)-net over F128, using
(81, 81+24, 2261693)-Net over F32 — Digital
Digital (81, 105, 2261693)-net over F32, using
(81, 81+24, large)-Net in Base 32 — Upper bound on s
There is no (81, 105, large)-net in base 32, because
- 22 times m-reduction [i] would yield (81, 83, large)-net in base 32, but