Best Known (27, 27+11, s)-Nets in Base 64
(27, 27+11, 54445)-Net over F64 — Constructive and digital
Digital (27, 38, 54445)-net over F64, using
- (u, u+v)-construction [i] based on
- 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
- 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 (2, 7, 2016)-net over F64, using
(27, 27+11, 419431)-Net in Base 64 — Constructive
(27, 38, 419431)-net in base 64, using
- net defined by OOA [i] based on OOA(6438, 419431, S64, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(6438, 2097156, S64, 11), using
- discarding factors based on OA(6438, 2097160, S64, 11), using
- discarding parts of the base [i] based on linear OA(12832, 2097160, F128, 11) (dual of [2097160, 2097128, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [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(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- discarding parts of the base [i] based on linear OA(12832, 2097160, F128, 11) (dual of [2097160, 2097128, 12]-code), using
- discarding factors based on OA(6438, 2097160, S64, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(6438, 2097156, S64, 11), using
(27, 27+11, 524957)-Net over F64 — Digital
Digital (27, 38, 524957)-net over F64, using
(27, 27+11, large)-Net in Base 64 — Upper bound on s
There is no (27, 38, large)-net in base 64, because
- 9 times m-reduction [i] would yield (27, 29, large)-net in base 64, but