Best Known (68−18, 68, s)-Nets in Base 32
(68−18, 68, 3740)-Net over F32 — Constructive and digital
Digital (50, 68, 3740)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 16, 99)-net over F32, using
- generalized (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 (0, 4, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 9, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 3, 33)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (34, 52, 3641)-net over F32, using
- net defined by OOA [i] based on linear OOA(3252, 3641, F32, 18, 18) (dual of [(3641, 18), 65486, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3252, 32769, F32, 18) (dual of [32769, 32717, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3252, 32771, F32, 18) (dual of [32771, 32719, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(3252, 32768, F32, 18) (dual of [32768, 32716, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(3249, 32768, F32, 17) (dual of [32768, 32719, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(320, 3, F32, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3252, 32771, F32, 18) (dual of [32771, 32719, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3252, 32769, F32, 18) (dual of [32769, 32717, 19]-code), using
- net defined by OOA [i] based on linear OOA(3252, 3641, F32, 18, 18) (dual of [(3641, 18), 65486, 19]-NRT-code), using
- digital (7, 16, 99)-net over F32, using
(68−18, 68, 29129)-Net in Base 32 — Constructive
(50, 68, 29129)-net in base 32, using
- net defined by OOA [i] based on OOA(3268, 29129, S32, 18, 18), using
- OA 9-folding and stacking [i] based on OA(3268, 262161, S32, 18), using
- discarding factors based on OA(3268, 262163, S32, 18), using
- discarding parts of the base [i] based on linear OA(6456, 262163, F64, 18) (dual of [262163, 262107, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(6452, 262144, F64, 18) (dual of [262144, 262092, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(644, 19, F64, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding parts of the base [i] based on linear OA(6456, 262163, F64, 18) (dual of [262163, 262107, 19]-code), using
- discarding factors based on OA(3268, 262163, S32, 18), using
- OA 9-folding and stacking [i] based on OA(3268, 262161, S32, 18), using
(68−18, 68, 242772)-Net over F32 — Digital
Digital (50, 68, 242772)-net over F32, using
(68−18, 68, large)-Net in Base 32 — Upper bound on s
There is no (50, 68, large)-net in base 32, because
- 16 times m-reduction [i] would yield (50, 52, large)-net in base 32, but