Best Known (45, 45+21, s)-Nets in Base 64
(45, 45+21, 26216)-Net over F64 — Constructive and digital
Digital (45, 66, 26216)-net over F64, using
- 641 times duplication [i] based on digital (44, 65, 26216)-net over F64, using
- net defined by OOA [i] based on linear OOA(6465, 26216, F64, 21, 21) (dual of [(26216, 21), 550471, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6465, 262161, F64, 21) (dual of [262161, 262096, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6465, 262163, F64, 21) (dual of [262163, 262098, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [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(6446, 262144, F64, 16) (dual of [262144, 262098, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(20) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(6465, 262163, F64, 21) (dual of [262163, 262098, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6465, 262161, F64, 21) (dual of [262161, 262096, 22]-code), using
- net defined by OOA [i] based on linear OOA(6465, 26216, F64, 21, 21) (dual of [(26216, 21), 550471, 22]-NRT-code), using
(45, 45+21, 190060)-Net over F64 — Digital
Digital (45, 66, 190060)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6466, 190060, F64, 21) (dual of [190060, 189994, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6466, 262168, F64, 21) (dual of [262168, 262102, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [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(6443, 262145, F64, 15) (dual of [262145, 262102, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(645, 23, F64, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6466, 262168, F64, 21) (dual of [262168, 262102, 22]-code), using
(45, 45+21, large)-Net in Base 64 — Upper bound on s
There is no (45, 66, large)-net in base 64, because
- 19 times m-reduction [i] would yield (45, 47, large)-net in base 64, but