Best Known (44, 55, s)-Nets in Base 64
(44, 55, 1808796)-Net over F64 — Constructive and digital
Digital (44, 55, 1808796)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (9, 14, 131076)-net over F64, using
- net defined by OOA [i] based on linear OOA(6414, 131076, F64, 5, 5) (dual of [(131076, 5), 655366, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(6414, 262153, F64, 5) (dual of [262153, 262139, 6]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(6413, 262145, F64, 5) (dual of [262145, 262132, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(647, 262145, F64, 3) (dual of [262145, 262138, 4]-code or 262145-cap in PG(6,64)), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(647, 8, F64, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,64)), using
- dual of repetition code with length 8 [i]
- linear OA(641, 8, F64, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, 64, F64, 1) (dual of [64, 63, 2]-code), using
- Reed–Solomon code RS(63,64) [i]
- discarding factors / shortening the dual code based on linear OA(641, 64, F64, 1) (dual of [64, 63, 2]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(6414, 262153, F64, 5) (dual of [262153, 262139, 6]-code), using
- net defined by OOA [i] based on linear OOA(6414, 131076, F64, 5, 5) (dual of [(131076, 5), 655366, 6]-NRT-code), using
- digital (30, 41, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- digital (9, 14, 131076)-net over F64, using
(44, 55, large)-Net over F64 — Digital
Digital (44, 55, large)-net over F64, using
- 4 times m-reduction [i] based on digital (44, 59, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6459, large, F64, 15) (dual of [large, large−59, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 2 times code embedding in larger space [i] based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6459, large, F64, 15) (dual of [large, large−59, 16]-code), using
(44, 55, large)-Net in Base 64 — Upper bound on s
There is no (44, 55, large)-net in base 64, because
- 9 times m-reduction [i] would yield (44, 46, large)-net in base 64, but