Best Known (88−23, 88, s)-Nets in Base 64
(88−23, 88, 24650)-Net over F64 — Constructive and digital
Digital (65, 88, 24650)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (10, 21, 819)-net over F64, using
- net defined by OOA [i] based on linear OOA(6421, 819, F64, 11, 11) (dual of [(819, 11), 8988, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- OOA 5-folding and stacking with additional row [i] based on linear OA(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using
- net defined by OOA [i] based on linear OOA(6421, 819, F64, 11, 11) (dual of [(819, 11), 8988, 12]-NRT-code), using
- digital (44, 67, 23831)-net over F64, using
- net defined by OOA [i] based on linear OOA(6467, 23831, F64, 23, 23) (dual of [(23831, 23), 548046, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6467, 262142, F64, 23) (dual of [262142, 262075, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(6467, 262144, F64, 23) (dual of [262144, 262077, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(6467, 262144, F64, 23) (dual of [262144, 262077, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6467, 262142, F64, 23) (dual of [262142, 262075, 24]-code), using
- net defined by OOA [i] based on linear OOA(6467, 23831, F64, 23, 23) (dual of [(23831, 23), 548046, 24]-NRT-code), using
- digital (10, 21, 819)-net over F64, using
(88−23, 88, 190653)-Net in Base 64 — Constructive
(65, 88, 190653)-net in base 64, using
- 641 times duplication [i] based on (64, 87, 190653)-net in base 64, using
- net defined by OOA [i] based on OOA(6487, 190653, S64, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(6487, 2097184, S64, 23), using
- discarding parts of the base [i] based on linear OA(12874, 2097184, F128, 23) (dual of [2097184, 2097110, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,7]) [i] based on
- linear OA(12867, 2097153, F128, 23) (dual of [2097153, 2097086, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- 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(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([0,11]) ⊂ C([0,7]) [i] based on
- discarding parts of the base [i] based on linear OA(12874, 2097184, F128, 23) (dual of [2097184, 2097110, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on OA(6487, 2097184, S64, 23), using
- net defined by OOA [i] based on OOA(6487, 190653, S64, 23, 23), using
(88−23, 88, 2411206)-Net over F64 — Digital
Digital (65, 88, 2411206)-net over F64, using
(88−23, 88, large)-Net in Base 64 — Upper bound on s
There is no (65, 88, large)-net in base 64, because
- 21 times m-reduction [i] would yield (65, 67, large)-net in base 64, but