Best Known (48, 48+23, s)-Nets in Base 64
(48, 48+23, 23832)-Net over F64 — Constructive and digital
Digital (48, 71, 23832)-net over F64, using
- 642 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
(48, 48+23, 144452)-Net over F64 — Digital
Digital (48, 71, 144452)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6471, 144452, F64, 23) (dual of [144452, 144381, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(6471, 262163, F64, 23) (dual of [262163, 262092, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [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(6452, 262144, F64, 18) (dual of [262144, 262092, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(644, 19, F64, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(6471, 262163, F64, 23) (dual of [262163, 262092, 24]-code), using
(48, 48+23, large)-Net in Base 64 — Upper bound on s
There is no (48, 71, large)-net in base 64, because
- 21 times m-reduction [i] would yield (48, 50, large)-net in base 64, but