Best Known (33, 33+16, s)-Nets in Base 64
(33, 33+16, 32769)-Net over F64 — Constructive and digital
Digital (33, 49, 32769)-net over F64, using
- 641 times duplication [i] based on digital (32, 48, 32769)-net over F64, using
- net defined by OOA [i] based on linear OOA(6448, 32769, F64, 16, 16) (dual of [(32769, 16), 524256, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(6448, 262152, F64, 16) (dual of [262152, 262104, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(6448, 262155, F64, 16) (dual of [262155, 262107, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- 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(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(642, 11, F64, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(6448, 262155, F64, 16) (dual of [262155, 262107, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(6448, 262152, F64, 16) (dual of [262152, 262104, 17]-code), using
- net defined by OOA [i] based on linear OOA(6448, 32769, F64, 16, 16) (dual of [(32769, 16), 524256, 17]-NRT-code), using
(33, 33+16, 149526)-Net over F64 — Digital
Digital (33, 49, 149526)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6449, 149526, F64, 16) (dual of [149526, 149477, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(6449, 262159, F64, 16) (dual of [262159, 262110, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- 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(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(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(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(6449, 262159, F64, 16) (dual of [262159, 262110, 17]-code), using
(33, 33+16, large)-Net in Base 64 — Upper bound on s
There is no (33, 49, large)-net in base 64, because
- 14 times m-reduction [i] would yield (33, 35, large)-net in base 64, but