Best Known (70−23, 70, s)-Nets in Base 64
(70−23, 70, 23832)-Net over F64 — Constructive and digital
Digital (47, 70, 23832)-net over F64, using
- 641 times duplication [i] based on digital (46, 69, 23832)-net over F64, using
- net defined by OOA [i] based on linear OOA(6469, 23832, F64, 23, 23) (dual of [(23832, 23), 548067, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6469, 262153, F64, 23) (dual of [262153, 262084, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(6469, 262155, F64, 23) (dual of [262155, 262086, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] 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]
- linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(6469, 262155, F64, 23) (dual of [262155, 262086, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6469, 262153, F64, 23) (dual of [262153, 262084, 24]-code), using
- net defined by OOA [i] based on linear OOA(6469, 23832, F64, 23, 23) (dual of [(23832, 23), 548067, 24]-NRT-code), using
(70−23, 70, 131080)-Net over F64 — Digital
Digital (47, 70, 131080)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6470, 131080, F64, 2, 23) (dual of [(131080, 2), 262090, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6470, 262160, F64, 23) (dual of [262160, 262090, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [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(6455, 262145, F64, 19) (dual of [262145, 262090, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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([0,11]) ⊂ C([0,9]) [i] based on
- OOA 2-folding [i] based on linear OA(6470, 262160, F64, 23) (dual of [262160, 262090, 24]-code), using
(70−23, 70, large)-Net in Base 64 — Upper bound on s
There is no (47, 70, large)-net in base 64, because
- 21 times m-reduction [i] would yield (47, 49, large)-net in base 64, but