Best Known (80−26, 80, s)-Nets in Base 64
(80−26, 80, 20166)-Net over F64 — Constructive and digital
Digital (54, 80, 20166)-net over F64, using
- 641 times duplication [i] based on digital (53, 79, 20166)-net over F64, using
- net defined by OOA [i] based on linear OOA(6479, 20166, F64, 26, 26) (dual of [(20166, 26), 524237, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(6479, 262158, F64, 26) (dual of [262158, 262079, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(6479, 262159, F64, 26) (dual of [262159, 262080, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(6476, 262144, F64, 26) (dual of [262144, 262068, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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(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(25) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(6479, 262159, F64, 26) (dual of [262159, 262080, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(6479, 262158, F64, 26) (dual of [262158, 262079, 27]-code), using
- net defined by OOA [i] based on linear OOA(6479, 20166, F64, 26, 26) (dual of [(20166, 26), 524237, 27]-NRT-code), using
(80−26, 80, 137192)-Net over F64 — Digital
Digital (54, 80, 137192)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6480, 137192, F64, 26) (dual of [137192, 137112, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(6480, 262163, F64, 26) (dual of [262163, 262083, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(20) [i] based on
- linear OA(6476, 262144, F64, 26) (dual of [262144, 262068, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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(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(25) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(6480, 262163, F64, 26) (dual of [262163, 262083, 27]-code), using
(80−26, 80, large)-Net in Base 64 — Upper bound on s
There is no (54, 80, large)-net in base 64, because
- 24 times m-reduction [i] would yield (54, 56, large)-net in base 64, but