Best Known (68−21, 68, s)-Nets in Base 64
(68−21, 68, 26217)-Net over F64 — Constructive and digital
Digital (47, 68, 26217)-net over F64, using
- 641 times duplication [i] based on digital (46, 67, 26217)-net over F64, using
- net defined by OOA [i] based on linear OOA(6467, 26217, F64, 21, 21) (dual of [(26217, 21), 550490, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6467, 262171, F64, 21) (dual of [262171, 262104, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- linear OA(6461, 262144, F64, 21) (dual of [262144, 262083, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(6440, 262144, F64, 14) (dual of [262144, 262104, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(646, 27, F64, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,64)), using
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- Reed–Solomon code RS(58,64) [i]
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(6467, 262171, F64, 21) (dual of [262171, 262104, 22]-code), using
- net defined by OOA [i] based on linear OOA(6467, 26217, F64, 21, 21) (dual of [(26217, 21), 550490, 22]-NRT-code), using
(68−21, 68, 262176)-Net over F64 — Digital
Digital (47, 68, 262176)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6468, 262176, F64, 21) (dual of [262176, 262108, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
- linear OA(6461, 262145, F64, 21) (dual of [262145, 262084, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(6437, 262145, F64, 13) (dual of [262145, 262108, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(647, 31, F64, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,64)), using
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- Reed–Solomon code RS(57,64) [i]
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
(68−21, 68, large)-Net in Base 64 — Upper bound on s
There is no (47, 68, large)-net in base 64, because
- 19 times m-reduction [i] would yield (47, 49, large)-net in base 64, but