Best Known (51−14, 51, s)-Nets in Base 32
(51−14, 51, 4758)-Net over F32 — Constructive and digital
Digital (37, 51, 4758)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 77)-net over F32, using
- (u, u+v)-construction [i] based on
- 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 (1, 8, 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, 3, 33)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (26, 40, 4681)-net over F32, using
- net defined by OOA [i] based on linear OOA(3240, 4681, F32, 14, 14) (dual of [(4681, 14), 65494, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(3240, 32767, F32, 14) (dual of [32767, 32727, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3240, 32768, F32, 14) (dual of [32768, 32728, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(3240, 32768, F32, 14) (dual of [32768, 32728, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(3240, 32767, F32, 14) (dual of [32767, 32727, 15]-code), using
- net defined by OOA [i] based on linear OOA(3240, 4681, F32, 14, 14) (dual of [(4681, 14), 65494, 15]-NRT-code), using
- digital (4, 11, 77)-net over F32, using
(51−14, 51, 37450)-Net in Base 32 — Constructive
(37, 51, 37450)-net in base 32, using
- 321 times duplication [i] based on (36, 50, 37450)-net in base 32, using
- net defined by OOA [i] based on OOA(3250, 37450, S32, 14, 14), using
- OA 7-folding and stacking [i] based on OA(3250, 262150, S32, 14), using
- discarding factors based on OA(3250, 262151, S32, 14), using
- discarding parts of the base [i] based on linear OA(6441, 262151, F64, 14) (dual of [262151, 262110, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(6440, 262144, F64, 14) (dual of [262144, 262104, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6434, 262144, F64, 12) (dual of [262144, 262110, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(6441, 262151, F64, 14) (dual of [262151, 262110, 15]-code), using
- discarding factors based on OA(3250, 262151, S32, 14), using
- OA 7-folding and stacking [i] based on OA(3250, 262150, S32, 14), using
- net defined by OOA [i] based on OOA(3250, 37450, S32, 14, 14), using
(51−14, 51, 146853)-Net over F32 — Digital
Digital (37, 51, 146853)-net over F32, using
(51−14, 51, large)-Net in Base 32 — Upper bound on s
There is no (37, 51, large)-net in base 32, because
- 12 times m-reduction [i] would yield (37, 39, large)-net in base 32, but