Best Known (70−17, 70, s)-Nets in Base 32
(70−17, 70, 131075)-Net over F32 — Constructive and digital
Digital (53, 70, 131075)-net over F32, using
- net defined by OOA [i] based on linear OOA(3270, 131075, F32, 17, 17) (dual of [(131075, 17), 2228205, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3270, 1048601, F32, 17) (dual of [1048601, 1048531, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3270, 1048606, F32, 17) (dual of [1048606, 1048536, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(3265, 1048577, F32, 17) (dual of [1048577, 1048512, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(3241, 1048577, F32, 11) (dual of [1048577, 1048536, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(325, 29, F32, 5) (dual of [29, 24, 6]-code or 29-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3270, 1048606, F32, 17) (dual of [1048606, 1048536, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3270, 1048601, F32, 17) (dual of [1048601, 1048531, 18]-code), using
(70−17, 70, 262144)-Net in Base 32 — Constructive
(53, 70, 262144)-net in base 32, using
- base change [i] based on digital (33, 50, 262144)-net over F128, using
- 1281 times duplication [i] based on digital (32, 49, 262144)-net over F128, using
- net defined by OOA [i] based on linear OOA(12849, 262144, F128, 17, 17) (dual of [(262144, 17), 4456399, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using
- net defined by OOA [i] based on linear OOA(12849, 262144, F128, 17, 17) (dual of [(262144, 17), 4456399, 18]-NRT-code), using
- 1281 times duplication [i] based on digital (32, 49, 262144)-net over F128, using
(70−17, 70, 1048606)-Net over F32 — Digital
Digital (53, 70, 1048606)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3270, 1048606, F32, 17) (dual of [1048606, 1048536, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(3265, 1048577, F32, 17) (dual of [1048577, 1048512, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(3241, 1048577, F32, 11) (dual of [1048577, 1048536, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(325, 29, F32, 5) (dual of [29, 24, 6]-code or 29-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
(70−17, 70, large)-Net in Base 32 — Upper bound on s
There is no (53, 70, large)-net in base 32, because
- 15 times m-reduction [i] would yield (53, 55, large)-net in base 32, but