Best Known (99−29, 99, s)-Nets in Base 32
(99−29, 99, 2373)-Net over F32 — Constructive and digital
Digital (70, 99, 2373)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 14, 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 (56, 85, 2340)-net over F32, using
- net defined by OOA [i] based on linear OOA(3285, 2340, F32, 29, 29) (dual of [(2340, 29), 67775, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3285, 32761, F32, 29) (dual of [32761, 32676, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3285, 32768, F32, 29) (dual of [32768, 32683, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(3285, 32768, F32, 29) (dual of [32768, 32683, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3285, 32761, F32, 29) (dual of [32761, 32676, 30]-code), using
- net defined by OOA [i] based on linear OOA(3285, 2340, F32, 29, 29) (dual of [(2340, 29), 67775, 30]-NRT-code), using
- digital (0, 14, 33)-net over F32, using
(99−29, 99, 4682)-Net in Base 32 — Constructive
(70, 99, 4682)-net in base 32, using
- 322 times duplication [i] based on (68, 97, 4682)-net in base 32, using
- net defined by OOA [i] based on OOA(3297, 4682, S32, 29, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(3297, 65549, S32, 29), using
- 1 times code embedding in larger space [i] based on OA(3296, 65548, S32, 29), using
- discarding parts of the base [i] based on linear OA(25660, 65548, F256, 29) (dual of [65548, 65488, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(25657, 65537, F256, 29) (dual of [65537, 65480, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- discarding parts of the base [i] based on linear OA(25660, 65548, F256, 29) (dual of [65548, 65488, 30]-code), using
- 1 times code embedding in larger space [i] based on OA(3296, 65548, S32, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(3297, 65549, S32, 29), using
- net defined by OOA [i] based on OOA(3297, 4682, S32, 29, 29), using
(99−29, 99, 76472)-Net over F32 — Digital
Digital (70, 99, 76472)-net over F32, using
(99−29, 99, large)-Net in Base 32 — Upper bound on s
There is no (70, 99, large)-net in base 32, because
- 27 times m-reduction [i] would yield (70, 72, large)-net in base 32, but