Best Known (41, 56, s)-Nets in Base 64
(41, 56, 38814)-Net over F64 — Constructive and digital
Digital (41, 56, 38814)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (6, 13, 1365)-net over F64, using
- net defined by OOA [i] based on linear OOA(6413, 1365, F64, 7, 7) (dual of [(1365, 7), 9542, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using
- net defined by OOA [i] based on linear OOA(6413, 1365, F64, 7, 7) (dual of [(1365, 7), 9542, 8]-NRT-code), using
- digital (28, 43, 37449)-net over F64, using
- net defined by OOA [i] based on linear OOA(6443, 37449, F64, 15, 15) (dual of [(37449, 15), 561692, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6443, 262144, F64, 15) (dual of [262144, 262101, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(6443, 262144, F64, 15) (dual of [262144, 262101, 16]-code), using
- net defined by OOA [i] based on linear OOA(6443, 37449, F64, 15, 15) (dual of [(37449, 15), 561692, 16]-NRT-code), using
- digital (6, 13, 1365)-net over F64, using
(41, 56, 299596)-Net in Base 64 — Constructive
(41, 56, 299596)-net in base 64, using
- base change [i] based on digital (33, 48, 299596)-net over F128, using
- net defined by OOA [i] based on linear OOA(12848, 299596, F128, 15, 15) (dual of [(299596, 15), 4493892, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12848, 2097173, F128, 15) (dual of [2097173, 2097125, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12848, 2097176, F128, 15) (dual of [2097176, 2097128, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- linear OA(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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(1285, 23, F128, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12848, 2097176, F128, 15) (dual of [2097176, 2097128, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12848, 2097173, F128, 15) (dual of [2097173, 2097125, 16]-code), using
- net defined by OOA [i] based on linear OOA(12848, 299596, F128, 15, 15) (dual of [(299596, 15), 4493892, 16]-NRT-code), using
(41, 56, 1610049)-Net over F64 — Digital
Digital (41, 56, 1610049)-net over F64, using
(41, 56, 1853428)-Net in Base 64
(41, 56, 1853428)-net in base 64, using
- base change [i] based on digital (33, 48, 1853428)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12848, 1853428, F128, 15) (dual of [1853428, 1853380, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12848, 2097176, F128, 15) (dual of [2097176, 2097128, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- linear OA(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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(1285, 23, F128, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12848, 2097176, F128, 15) (dual of [2097176, 2097128, 16]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12848, 1853428, F128, 15) (dual of [1853428, 1853380, 16]-code), using
(41, 56, large)-Net in Base 64 — Upper bound on s
There is no (41, 56, large)-net in base 64, because
- 13 times m-reduction [i] would yield (41, 43, large)-net in base 64, but