Best Known (45, 68, s)-Nets in Base 64
(45, 68, 23831)-Net over F64 — Constructive and digital
Digital (45, 68, 23831)-net over F64, using
- 641 times duplication [i] based on digital (44, 67, 23831)-net over F64, using
- net defined by OOA [i] based on linear OOA(6467, 23831, F64, 23, 23) (dual of [(23831, 23), 548046, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6467, 262142, F64, 23) (dual of [262142, 262075, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(6467, 262144, F64, 23) (dual of [262144, 262077, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(6467, 262144, F64, 23) (dual of [262144, 262077, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6467, 262142, F64, 23) (dual of [262142, 262075, 24]-code), using
- net defined by OOA [i] based on linear OOA(6467, 23831, F64, 23, 23) (dual of [(23831, 23), 548046, 24]-NRT-code), using
(45, 68, 120317)-Net over F64 — Digital
Digital (45, 68, 120317)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6468, 120317, F64, 2, 23) (dual of [(120317, 2), 240566, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6468, 131076, F64, 2, 23) (dual of [(131076, 2), 262084, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6468, 262152, F64, 23) (dual of [262152, 262084, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(6467, 262145, F64, 23) (dual of [262145, 262078, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- OOA 2-folding [i] based on linear OA(6468, 262152, F64, 23) (dual of [262152, 262084, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(6468, 131076, F64, 2, 23) (dual of [(131076, 2), 262084, 24]-NRT-code), using
(45, 68, large)-Net in Base 64 — Upper bound on s
There is no (45, 68, large)-net in base 64, because
- 21 times m-reduction [i] would yield (45, 47, large)-net in base 64, but