Best Known (70, 70+35, s)-Nets in Base 32
(70, 70+35, 1928)-Net over F32 — Constructive and digital
Digital (70, 105, 1928)-net over F32, using
- 322 times duplication [i] based on digital (68, 103, 1928)-net over F32, using
- net defined by OOA [i] based on linear OOA(32103, 1928, F32, 35, 35) (dual of [(1928, 35), 67377, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(32103, 32777, F32, 35) (dual of [32777, 32674, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(32103, 32780, F32, 35) (dual of [32780, 32677, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(32100, 32768, F32, 35) (dual of [32768, 32668, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3291, 32768, F32, 31) (dual of [32768, 32677, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(323, 12, F32, 3) (dual of [12, 9, 4]-code or 12-arc in PG(2,32) or 12-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(32103, 32780, F32, 35) (dual of [32780, 32677, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(32103, 32777, F32, 35) (dual of [32777, 32674, 36]-code), using
- net defined by OOA [i] based on linear OOA(32103, 1928, F32, 35, 35) (dual of [(1928, 35), 67377, 36]-NRT-code), using
(70, 70+35, 23507)-Net over F32 — Digital
Digital (70, 105, 23507)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32105, 23507, F32, 35) (dual of [23507, 23402, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(32105, 32788, F32, 35) (dual of [32788, 32683, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(32100, 32768, F32, 35) (dual of [32768, 32668, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- 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]
- linear OA(325, 20, F32, 5) (dual of [20, 15, 6]-code or 20-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 Ce(34) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(32105, 32788, F32, 35) (dual of [32788, 32683, 36]-code), using
(70, 70+35, large)-Net in Base 32 — Upper bound on s
There is no (70, 105, large)-net in base 32, because
- 33 times m-reduction [i] would yield (70, 72, large)-net in base 32, but