Best Known (29, 40, s)-Nets in Base 64
(29, 40, 54510)-Net over F64 — Constructive and digital
Digital (29, 40, 54510)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (4, 9, 2081)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 2, 65)-net over F64, using
- digital (2, 7, 2016)-net over F64, using
- net defined by OOA [i] based on linear OOA(647, 2016, F64, 5, 5) (dual of [(2016, 5), 10073, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- net defined by OOA [i] based on linear OOA(647, 2016, F64, 5, 5) (dual of [(2016, 5), 10073, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- digital (20, 31, 52429)-net over F64, using
- net defined by OOA [i] based on linear OOA(6431, 52429, F64, 11, 11) (dual of [(52429, 11), 576688, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6431, 262146, F64, 11) (dual of [262146, 262115, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6431, 262147, F64, 11) (dual of [262147, 262116, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(6431, 262144, F64, 11) (dual of [262144, 262113, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(6431, 262147, F64, 11) (dual of [262147, 262116, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6431, 262146, F64, 11) (dual of [262146, 262115, 12]-code), using
- net defined by OOA [i] based on linear OOA(6431, 52429, F64, 11, 11) (dual of [(52429, 11), 576688, 12]-NRT-code), using
- digital (4, 9, 2081)-net over F64, using
(29, 40, 419433)-Net in Base 64 — Constructive
(29, 40, 419433)-net in base 64, using
- net defined by OOA [i] based on OOA(6440, 419433, S64, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(6440, 2097166, S64, 11), using
- discarding factors based on OA(6440, 2097168, S64, 11), using
- discarding parts of the base [i] based on linear OA(12834, 2097168, F128, 11) (dual of [2097168, 2097134, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- linear OA(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(12819, 2097153, F128, 7) (dual of [2097153, 2097134, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- discarding parts of the base [i] based on linear OA(12834, 2097168, F128, 11) (dual of [2097168, 2097134, 12]-code), using
- discarding factors based on OA(6440, 2097168, S64, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(6440, 2097166, S64, 11), using
(29, 40, 1206028)-Net over F64 — Digital
Digital (29, 40, 1206028)-net over F64, using
(29, 40, large)-Net in Base 64 — Upper bound on s
There is no (29, 40, large)-net in base 64, because
- 9 times m-reduction [i] would yield (29, 31, large)-net in base 64, but