Best Known (18, 18+14, s)-Nets in Base 64
(18, 18+14, 587)-Net over F64 — Constructive and digital
Digital (18, 32, 587)-net over F64, using
- 1 times m-reduction [i] based on digital (18, 33, 587)-net over F64, using
- net defined by OOA [i] based on linear OOA(6433, 587, F64, 15, 15) (dual of [(587, 15), 8772, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6433, 4110, F64, 15) (dual of [4110, 4077, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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]
- linear OA(644, 14, F64, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(6433, 4110, F64, 15) (dual of [4110, 4077, 16]-code), using
- net defined by OOA [i] based on linear OOA(6433, 587, F64, 15, 15) (dual of [(587, 15), 8772, 16]-NRT-code), using
(18, 18+14, 2340)-Net in Base 64 — Constructive
(18, 32, 2340)-net in base 64, using
- net defined by OOA [i] based on OOA(6432, 2340, S64, 14, 14), using
- OA 7-folding and stacking [i] based on OA(6432, 16380, S64, 14), using
- discarding factors based on OA(6432, 16386, S64, 14), using
- discarding parts of the base [i] based on linear OA(12827, 16386, F128, 14) (dual of [16386, 16359, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(12827, 16384, F128, 14) (dual of [16384, 16357, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- discarding parts of the base [i] based on linear OA(12827, 16386, F128, 14) (dual of [16386, 16359, 15]-code), using
- discarding factors based on OA(6432, 16386, S64, 14), using
- OA 7-folding and stacking [i] based on OA(6432, 16380, S64, 14), using
(18, 18+14, 3885)-Net over F64 — Digital
Digital (18, 32, 3885)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6432, 3885, F64, 14) (dual of [3885, 3853, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6432, 4113, F64, 14) (dual of [4113, 4081, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(7) [i] based on
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to Ce(13) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(6432, 4113, F64, 14) (dual of [4113, 4081, 15]-code), using
(18, 18+14, large)-Net in Base 64 — Upper bound on s
There is no (18, 32, large)-net in base 64, because
- 12 times m-reduction [i] would yield (18, 20, large)-net in base 64, but