Best Known (80−21, 80, s)-Nets in Base 64
(80−21, 80, 27033)-Net over F64 — Constructive and digital
Digital (59, 80, 27033)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (9, 19, 819)-net over F64, using
- net defined by OOA [i] based on linear OOA(6419, 819, F64, 10, 10) (dual of [(819, 10), 8171, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6419, 4095, F64, 10) (dual of [4095, 4076, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6419, 4095, F64, 10) (dual of [4095, 4076, 11]-code), using
- net defined by OOA [i] based on linear OOA(6419, 819, F64, 10, 10) (dual of [(819, 10), 8171, 11]-NRT-code), using
- digital (40, 61, 26214)-net over F64, using
- net defined by OOA [i] based on linear OOA(6461, 26214, F64, 21, 21) (dual of [(26214, 21), 550433, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6461, 262141, F64, 21) (dual of [262141, 262080, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6461, 262144, F64, 21) (dual of [262144, 262083, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(6461, 262144, F64, 21) (dual of [262144, 262083, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6461, 262141, F64, 21) (dual of [262141, 262080, 22]-code), using
- net defined by OOA [i] based on linear OOA(6461, 26214, F64, 21, 21) (dual of [(26214, 21), 550433, 22]-NRT-code), using
- digital (9, 19, 819)-net over F64, using
(80−21, 80, 209718)-Net in Base 64 — Constructive
(59, 80, 209718)-net in base 64, using
- net defined by OOA [i] based on OOA(6480, 209718, S64, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(6480, 2097181, S64, 21), using
- discarding factors based on OA(6480, 2097184, S64, 21), using
- discarding parts of the base [i] based on linear OA(12868, 2097184, F128, 21) (dual of [2097184, 2097116, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
- linear OA(12861, 2097153, F128, 21) (dual of [2097153, 2097092, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1287, 31, F128, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
- discarding parts of the base [i] based on linear OA(12868, 2097184, F128, 21) (dual of [2097184, 2097116, 22]-code), using
- discarding factors based on OA(6480, 2097184, S64, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(6480, 2097181, S64, 21), using
(80−21, 80, 2211503)-Net over F64 — Digital
Digital (59, 80, 2211503)-net over F64, using
(80−21, 80, large)-Net in Base 64 — Upper bound on s
There is no (59, 80, large)-net in base 64, because
- 19 times m-reduction [i] would yield (59, 61, large)-net in base 64, but