Best Known (68−22, 68, s)-Nets in Base 32
(68−22, 68, 2980)-Net over F32 — Constructive and digital
Digital (46, 68, 2980)-net over F32, using
- 321 times duplication [i] based on digital (45, 67, 2980)-net over F32, using
- net defined by OOA [i] based on linear OOA(3267, 2980, F32, 22, 22) (dual of [(2980, 22), 65493, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3267, 32780, F32, 22) (dual of [32780, 32713, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3267, 32783, F32, 22) (dual of [32783, 32716, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(3264, 32768, F32, 22) (dual of [32768, 32704, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(323, 15, F32, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,32) or 15-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(21) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(3267, 32783, F32, 22) (dual of [32783, 32716, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3267, 32780, F32, 22) (dual of [32780, 32713, 23]-code), using
- net defined by OOA [i] based on linear OOA(3267, 2980, F32, 22, 22) (dual of [(2980, 22), 65493, 23]-NRT-code), using
(68−22, 68, 29516)-Net over F32 — Digital
Digital (46, 68, 29516)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3268, 29516, F32, 22) (dual of [29516, 29448, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3268, 32787, F32, 22) (dual of [32787, 32719, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3264, 32768, F32, 22) (dual of [32768, 32704, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(324, 19, F32, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3268, 32787, F32, 22) (dual of [32787, 32719, 23]-code), using
(68−22, 68, large)-Net in Base 32 — Upper bound on s
There is no (46, 68, large)-net in base 32, because
- 20 times m-reduction [i] would yield (46, 48, large)-net in base 32, but