Best Known (23, 23+9, s)-Nets in Base 64
(23, 23+9, 67585)-Net over F64 — Constructive and digital
Digital (23, 32, 67585)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 7, 2049)-net over F64, using
- net defined by OOA [i] based on linear OOA(647, 2049, F64, 4, 4) (dual of [(2049, 4), 8189, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(647, 2049, F64, 3, 4) (dual of [(2049, 3), 6140, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(647, 4096, F64, 4) (dual of [4096, 4089, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(645, 4096, F64, 3) (dual of [4096, 4091, 4]-code or 4096-cap in PG(4,64)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 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(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- appending kth column [i] based on linear OOA(647, 2049, F64, 3, 4) (dual of [(2049, 3), 6140, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(647, 2049, F64, 4, 4) (dual of [(2049, 4), 8189, 5]-NRT-code), using
- digital (16, 25, 65536)-net over F64, using
- net defined by OOA [i] based on linear OOA(6425, 65536, F64, 9, 9) (dual of [(65536, 9), 589799, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6425, 262145, F64, 9) (dual of [262145, 262120, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(6425, 262145, F64, 9) (dual of [262145, 262120, 10]-code), using
- net defined by OOA [i] based on linear OOA(6425, 65536, F64, 9, 9) (dual of [(65536, 9), 589799, 10]-NRT-code), using
- digital (3, 7, 2049)-net over F64, using
(23, 23+9, 524290)-Net in Base 64 — Constructive
(23, 32, 524290)-net in base 64, using
- net defined by OOA [i] based on OOA(6432, 524290, S64, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(6432, 2097161, S64, 9), using
- discarding factors based on OA(6432, 2097163, S64, 9), using
- discarding parts of the base [i] based on linear OA(12827, 2097163, F128, 9) (dual of [2097163, 2097136, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(12825, 2097152, F128, 9) (dual of [2097152, 2097127, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(12827, 2097163, F128, 9) (dual of [2097163, 2097136, 10]-code), using
- discarding factors based on OA(6432, 2097163, S64, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(6432, 2097161, S64, 9), using
(23, 23+9, 1002469)-Net over F64 — Digital
Digital (23, 32, 1002469)-net over F64, using
(23, 23+9, large)-Net in Base 64 — Upper bound on s
There is no (23, 32, large)-net in base 64, because
- 7 times m-reduction [i] would yield (23, 25, large)-net in base 64, but