Best Known (27−12, 27, s)-Nets in Base 64
(27−12, 27, 685)-Net over F64 — Constructive and digital
Digital (15, 27, 685)-net over F64, using
- net defined by OOA [i] based on linear OOA(6427, 685, F64, 12, 12) (dual of [(685, 12), 8193, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6427, 4110, F64, 12) (dual of [4110, 4083, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- 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]
- 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(11) ⊂ Ce(6) [i] based on
- OA 6-folding and stacking [i] based on linear OA(6427, 4110, F64, 12) (dual of [4110, 4083, 13]-code), using
(27−12, 27, 2731)-Net in Base 64 — Constructive
(15, 27, 2731)-net in base 64, using
- net defined by OOA [i] based on OOA(6427, 2731, S64, 12, 12), using
- OA 6-folding and stacking [i] based on OA(6427, 16386, S64, 12), using
- discarding parts of the base [i] based on linear OA(12823, 16386, F128, 12) (dual of [16386, 16363, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12821, 16384, F128, 11) (dual of [16384, 16363, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- discarding parts of the base [i] based on linear OA(12823, 16386, F128, 12) (dual of [16386, 16363, 13]-code), using
- OA 6-folding and stacking [i] based on OA(6427, 16386, S64, 12), using
(27−12, 27, 3566)-Net over F64 — Digital
Digital (15, 27, 3566)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6427, 3566, F64, 12) (dual of [3566, 3539, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6427, 4110, F64, 12) (dual of [4110, 4083, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- 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]
- 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(11) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(6427, 4110, F64, 12) (dual of [4110, 4083, 13]-code), using
(27−12, 27, 6378098)-Net in Base 64 — Upper bound on s
There is no (15, 27, 6378099)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 5 846007 883680 238348 460591 952185 953980 450835 113628 > 6427 [i]