Best Known (64, 64+18, s)-Nets in Base 32
(64, 64+18, 116574)-Net over F32 — Constructive and digital
Digital (64, 82, 116574)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 66)-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 (0, 9, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 4, 33)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (51, 69, 116508)-net over F32, using
- net defined by OOA [i] based on linear OOA(3269, 116508, F32, 18, 18) (dual of [(116508, 18), 2097075, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3269, 1048572, F32, 18) (dual of [1048572, 1048503, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3269, 1048576, F32, 18) (dual of [1048576, 1048507, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(3269, 1048576, F32, 18) (dual of [1048576, 1048507, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3269, 1048572, F32, 18) (dual of [1048572, 1048503, 19]-code), using
- net defined by OOA [i] based on linear OOA(3269, 116508, F32, 18, 18) (dual of [(116508, 18), 2097075, 19]-NRT-code), using
- digital (4, 13, 66)-net over F32, using
(64, 64+18, 233050)-Net in Base 32 — Constructive
(64, 82, 233050)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 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
- (55, 73, 233017)-net in base 32, using
- net defined by OOA [i] based on OOA(3273, 233017, S32, 18, 18), using
- OA 9-folding and stacking [i] based on OA(3273, 2097153, S32, 18), using
- discarding factors based on OA(3273, 2097155, S32, 18), using
- discarding parts of the base [i] based on linear OA(12852, 2097155, F128, 18) (dual of [2097155, 2097103, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(12849, 2097152, F128, 17) (dual of [2097152, 2097103, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- discarding parts of the base [i] based on linear OA(12852, 2097155, F128, 18) (dual of [2097155, 2097103, 19]-code), using
- discarding factors based on OA(3273, 2097155, S32, 18), using
- OA 9-folding and stacking [i] based on OA(3273, 2097153, S32, 18), using
- net defined by OOA [i] based on OOA(3273, 233017, S32, 18, 18), using
- digital (0, 9, 33)-net over F32, using
(64, 64+18, 4214243)-Net over F32 — Digital
Digital (64, 82, 4214243)-net over F32, using
(64, 64+18, large)-Net in Base 32 — Upper bound on s
There is no (64, 82, large)-net in base 32, because
- 16 times m-reduction [i] would yield (64, 66, large)-net in base 32, but