Best Known (7, 15, s)-Nets in Base 64
(7, 15, 1024)-Net over F64 — Constructive and digital
Digital (7, 15, 1024)-net over F64, using
- net defined by OOA [i] based on linear OOA(6415, 1024, F64, 8, 8) (dual of [(1024, 8), 8177, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on 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]
- OA 4-folding and stacking [i] based on linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using
(7, 15, 2049)-Net over F64 — Digital
Digital (7, 15, 2049)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6415, 2049, F64, 2, 8) (dual of [(2049, 2), 4083, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6415, 4098, F64, 8) (dual of [4098, 4083, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- 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(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(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(7) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(6415, 4098, F64, 8) (dual of [4098, 4083, 9]-code), using
(7, 15, 208393)-Net in Base 64 — Upper bound on s
There is no (7, 15, 208394)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 1237 954926 337249 774049 331454 > 6415 [i]