Best Known (68−22, 68, s)-Nets in Base 64
(68−22, 68, 23833)-Net over F64 — Constructive and digital
Digital (46, 68, 23833)-net over F64, using
- net defined by OOA [i] based on linear OOA(6468, 23833, F64, 22, 22) (dual of [(23833, 22), 524258, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(6468, 262163, F64, 22) (dual of [262163, 262095, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(6449, 262144, F64, 17) (dual of [262144, 262095, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(21) ⊂ Ce(16) [i] based on
- OA 11-folding and stacking [i] based on linear OA(6468, 262163, F64, 22) (dual of [262163, 262095, 23]-code), using
(68−22, 68, 148129)-Net over F64 — Digital
Digital (46, 68, 148129)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6468, 148129, F64, 22) (dual of [148129, 148061, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(6468, 262163, F64, 22) (dual of [262163, 262095, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(6449, 262144, F64, 17) (dual of [262144, 262095, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(6468, 262163, F64, 22) (dual of [262163, 262095, 23]-code), using
(68−22, 68, large)-Net in Base 64 — Upper bound on s
There is no (46, 68, large)-net in base 64, because
- 20 times m-reduction [i] would yield (46, 48, large)-net in base 64, but