Best Known (53, 77, s)-Nets in Base 64
(53, 77, 21847)-Net over F64 — Constructive and digital
Digital (53, 77, 21847)-net over F64, using
- 1 times m-reduction [i] based on digital (53, 78, 21847)-net over F64, using
- net defined by OOA [i] based on linear OOA(6478, 21847, F64, 25, 25) (dual of [(21847, 25), 546097, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(6478, 262165, F64, 25) (dual of [262165, 262087, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(6478, 262168, F64, 25) (dual of [262168, 262090, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(6473, 262145, F64, 25) (dual of [262145, 262072, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(6455, 262145, F64, 19) (dual of [262145, 262090, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6478, 262168, F64, 25) (dual of [262168, 262090, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(6478, 262165, F64, 25) (dual of [262165, 262087, 26]-code), using
- net defined by OOA [i] based on linear OOA(6478, 21847, F64, 25, 25) (dual of [(21847, 25), 546097, 26]-NRT-code), using
(53, 77, 249474)-Net over F64 — Digital
Digital (53, 77, 249474)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6477, 249474, F64, 24) (dual of [249474, 249397, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(6477, 262175, F64, 24) (dual of [262175, 262098, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(15) [i] based on
- linear OA(6470, 262144, F64, 24) (dual of [262144, 262074, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(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 Ce(23) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(6477, 262175, F64, 24) (dual of [262175, 262098, 25]-code), using
(53, 77, large)-Net in Base 64 — Upper bound on s
There is no (53, 77, large)-net in base 64, because
- 22 times m-reduction [i] would yield (53, 55, large)-net in base 64, but