Best Known (24−7, 24, s)-Nets in Base 64
(24−7, 24, 95647)-Net over F64 — Constructive and digital
Digital (17, 24, 95647)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (2, 5, 8265)-net over F64, using
- net defined by OOA [i] based on linear OOA(645, 8265, F64, 3, 3) (dual of [(8265, 3), 24790, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(645, 8265, F64, 2, 3) (dual of [(8265, 2), 16525, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(645, 8265, F64, 3, 3) (dual of [(8265, 3), 24790, 4]-NRT-code), using
- digital (12, 19, 87382)-net over F64, using
- net defined by OOA [i] based on linear OOA(6419, 87382, F64, 7, 7) (dual of [(87382, 7), 611655, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(6419, 262147, F64, 7) (dual of [262147, 262128, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(6419, 262144, F64, 7) (dual of [262144, 262125, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(6416, 262144, F64, 6) (dual of [262144, 262128, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(6419, 262147, F64, 7) (dual of [262147, 262128, 8]-code), using
- net defined by OOA [i] based on linear OOA(6419, 87382, F64, 7, 7) (dual of [(87382, 7), 611655, 8]-NRT-code), using
- digital (2, 5, 8265)-net over F64, using
(24−7, 24, 699053)-Net in Base 64 — Constructive
(17, 24, 699053)-net in base 64, using
- net defined by OOA [i] based on OOA(6424, 699053, S64, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(6424, 2097160, S64, 7), using
- discarding factors based on OA(6424, 2097161, S64, 7), using
- discarding parts of the base [i] based on linear OA(12820, 2097161, F128, 7) (dual of [2097161, 2097141, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- 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(12813, 2097153, F128, 5) (dual of [2097153, 2097140, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(1287, 8, F128, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,128)), using
- dual of repetition code with length 8 [i]
- linear OA(1281, 8, F128, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- Reed–Solomon code RS(127,128) [i]
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- discarding parts of the base [i] based on linear OA(12820, 2097161, F128, 7) (dual of [2097161, 2097141, 8]-code), using
- discarding factors based on OA(6424, 2097161, S64, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(6424, 2097160, S64, 7), using
(24−7, 24, 797266)-Net over F64 — Digital
Digital (17, 24, 797266)-net over F64, using
(24−7, 24, 2086555)-Net in Base 64
(17, 24, 2086555)-net in base 64, using
- net defined by OOA [i] based on OOA(6424, 2086555, S64, 9, 7), using
- OOA stacking with additional row [i] based on OOA(6424, 2086556, S64, 3, 7), using
- discarding parts of the base [i] based on linear OOA(12820, 2086556, F128, 3, 7) (dual of [(2086556, 3), 6259648, 8]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12820, 2086556, F128, 7) (dual of [2086556, 2086536, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(12820, 2097160, F128, 7) (dual of [2097160, 2097140, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- 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(12813, 2097153, F128, 5) (dual of [2097153, 2097140, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (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([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12820, 2097160, F128, 7) (dual of [2097160, 2097140, 8]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12820, 2086556, F128, 7) (dual of [2086556, 2086536, 8]-code), using
- discarding parts of the base [i] based on linear OOA(12820, 2086556, F128, 3, 7) (dual of [(2086556, 3), 6259648, 8]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(6424, 2086556, S64, 3, 7), using
(24−7, 24, large)-Net in Base 64 — Upper bound on s
There is no (17, 24, large)-net in base 64, because
- 5 times m-reduction [i] would yield (17, 19, large)-net in base 64, but