Best Known (72, 72+17, s)-Nets in Base 64
(72, 72+17, 1114113)-Net over F64 — Constructive and digital
Digital (72, 89, 1114113)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (16, 24, 65538)-net over F64, using
- net defined by OOA [i] based on linear OOA(6424, 65538, F64, 8, 8) (dual of [(65538, 8), 524280, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(6424, 262152, F64, 8) (dual of [262152, 262128, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(6424, 262155, F64, 8) (dual of [262155, 262131, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(6422, 262144, F64, 8) (dual of [262144, 262122, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(6413, 262144, F64, 5) (dual of [262144, 262131, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(6424, 262155, F64, 8) (dual of [262155, 262131, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(6424, 262152, F64, 8) (dual of [262152, 262128, 9]-code), using
- net defined by OOA [i] based on linear OOA(6424, 65538, F64, 8, 8) (dual of [(65538, 8), 524280, 9]-NRT-code), using
- digital (48, 65, 1048575)-net over F64, using
- net defined by OOA [i] based on linear OOA(6465, 1048575, F64, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6465, 8388601, F64, 17) (dual of [8388601, 8388536, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6465, 8388601, F64, 17) (dual of [8388601, 8388536, 18]-code), using
- net defined by OOA [i] based on linear OOA(6465, 1048575, F64, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- digital (16, 24, 65538)-net over F64, using
(72, 72+17, large)-Net over F64 — Digital
Digital (72, 89, large)-net over F64, using
- 641 times duplication [i] based on digital (71, 88, large)-net over F64, using
- t-expansion [i] based on digital (66, 88, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- t-expansion [i] based on digital (66, 88, large)-net over F64, using
(72, 72+17, large)-Net in Base 64 — Upper bound on s
There is no (72, 89, large)-net in base 64, because
- 15 times m-reduction [i] would yield (72, 74, large)-net in base 64, but