Best Known (55, 55+25, s)-Nets in Base 64
(55, 55+25, 21847)-Net over F64 — Constructive and digital
Digital (55, 80, 21847)-net over F64, using
- 642 times duplication [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
(55, 55+25, 239294)-Net over F64 — Digital
Digital (55, 80, 239294)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6480, 239294, F64, 25) (dual of [239294, 239214, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(6480, 262176, F64, 25) (dual of [262176, 262096, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [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(6449, 262145, F64, 17) (dual of [262145, 262096, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,12]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6480, 262176, F64, 25) (dual of [262176, 262096, 26]-code), using
(55, 55+25, large)-Net in Base 64 — Upper bound on s
There is no (55, 80, large)-net in base 64, because
- 23 times m-reduction [i] would yield (55, 57, large)-net in base 64, but