Best Known (37, 54, s)-Nets in Base 64
(37, 54, 32770)-Net over F64 — Constructive and digital
Digital (37, 54, 32770)-net over F64, using
- 641 times duplication [i] based on digital (36, 53, 32770)-net over F64, using
- net defined by OOA [i] based on linear OOA(6453, 32770, F64, 17, 17) (dual of [(32770, 17), 557037, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6453, 262161, F64, 17) (dual of [262161, 262108, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(6453, 262163, F64, 17) (dual of [262163, 262110, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(6449, 262144, F64, 17) (dual of [262144, 262095, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(6434, 262144, F64, 12) (dual of [262144, 262110, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(16) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(6453, 262163, F64, 17) (dual of [262163, 262110, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6453, 262161, F64, 17) (dual of [262161, 262108, 18]-code), using
- net defined by OOA [i] based on linear OOA(6453, 32770, F64, 17, 17) (dual of [(32770, 17), 557037, 18]-NRT-code), using
(37, 54, 245612)-Net over F64 — Digital
Digital (37, 54, 245612)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6454, 245612, F64, 17) (dual of [245612, 245558, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(6454, 262168, F64, 17) (dual of [262168, 262114, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- 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(6431, 262145, F64, 11) (dual of [262145, 262114, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6454, 262168, F64, 17) (dual of [262168, 262114, 18]-code), using
(37, 54, large)-Net in Base 64 — Upper bound on s
There is no (37, 54, large)-net in base 64, because
- 15 times m-reduction [i] would yield (37, 39, large)-net in base 64, but