Best Known (45, 67, s)-Nets in Base 64
(45, 67, 23832)-Net over F64 — Constructive and digital
Digital (45, 67, 23832)-net over F64, using
- 641 times duplication [i] based on digital (44, 66, 23832)-net over F64, using
- net defined by OOA [i] based on linear OOA(6466, 23832, F64, 22, 22) (dual of [(23832, 22), 524238, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(6466, 262152, F64, 22) (dual of [262152, 262086, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(6466, 262155, F64, 22) (dual of [262155, 262089, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [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(6455, 262144, F64, 19) (dual of [262144, 262089, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(642, 11, F64, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(6466, 262155, F64, 22) (dual of [262155, 262089, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(6466, 262152, F64, 22) (dual of [262152, 262086, 23]-code), using
- net defined by OOA [i] based on linear OOA(6466, 23832, F64, 22, 22) (dual of [(23832, 22), 524238, 23]-NRT-code), using
(45, 67, 131079)-Net over F64 — Digital
Digital (45, 67, 131079)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6467, 131079, F64, 2, 22) (dual of [(131079, 2), 262091, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6467, 262158, F64, 22) (dual of [262158, 262091, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(6467, 262159, F64, 22) (dual of [262159, 262092, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [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(6452, 262144, F64, 18) (dual of [262144, 262092, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(643, 15, F64, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,64) or 15-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(6467, 262159, F64, 22) (dual of [262159, 262092, 23]-code), using
- OOA 2-folding [i] based on linear OA(6467, 262158, F64, 22) (dual of [262158, 262091, 23]-code), using
(45, 67, large)-Net in Base 64 — Upper bound on s
There is no (45, 67, large)-net in base 64, because
- 20 times m-reduction [i] would yield (45, 47, large)-net in base 64, but