Best Known (76−25, 76, s)-Nets in Base 64
(76−25, 76, 21846)-Net over F64 — Constructive and digital
Digital (51, 76, 21846)-net over F64, using
- 641 times duplication [i] based on digital (50, 75, 21846)-net over F64, using
- net defined by OOA [i] based on linear OOA(6475, 21846, F64, 25, 25) (dual of [(21846, 25), 546075, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(6475, 262153, F64, 25) (dual of [262153, 262078, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(6475, 262155, F64, 25) (dual of [262155, 262080, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(6473, 262144, F64, 25) (dual of [262144, 262071, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(6475, 262155, F64, 25) (dual of [262155, 262080, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(6475, 262153, F64, 25) (dual of [262153, 262078, 26]-code), using
- net defined by OOA [i] based on linear OOA(6475, 21846, F64, 25, 25) (dual of [(21846, 25), 546075, 26]-NRT-code), using
(76−25, 76, 131080)-Net over F64 — Digital
Digital (51, 76, 131080)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6476, 131080, F64, 2, 25) (dual of [(131080, 2), 262084, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6476, 262160, F64, 25) (dual of [262160, 262084, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [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(6461, 262145, F64, 21) (dual of [262145, 262084, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [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 C([0,12]) ⊂ C([0,10]) [i] based on
- OOA 2-folding [i] based on linear OA(6476, 262160, F64, 25) (dual of [262160, 262084, 26]-code), using
(76−25, 76, large)-Net in Base 64 — Upper bound on s
There is no (51, 76, large)-net in base 64, because
- 23 times m-reduction [i] would yield (51, 53, large)-net in base 64, but