Best Known (35−18, 35, s)-Nets in Base 64
(35−18, 35, 455)-Net over F64 — Constructive and digital
Digital (17, 35, 455)-net over F64, using
- net defined by OOA [i] based on linear OOA(6435, 455, F64, 18, 18) (dual of [(455, 18), 8155, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6435, 4095, F64, 18) (dual of [4095, 4060, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(6435, 4095, F64, 18) (dual of [4095, 4060, 19]-code), using
(35−18, 35, 1284)-Net over F64 — Digital
Digital (17, 35, 1284)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6435, 1284, F64, 3, 18) (dual of [(1284, 3), 3817, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6435, 1366, F64, 3, 18) (dual of [(1366, 3), 4063, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6435, 4098, F64, 18) (dual of [4098, 4063, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(6433, 4096, F64, 17) (dual of [4096, 4063, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- OOA 3-folding [i] based on linear OA(6435, 4098, F64, 18) (dual of [4098, 4063, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(6435, 1366, F64, 3, 18) (dual of [(1366, 3), 4063, 19]-NRT-code), using
(35−18, 35, 695731)-Net in Base 64 — Upper bound on s
There is no (17, 35, 695732)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 1645 519368 427483 255773 576799 749560 904829 687507 077075 436769 996338 > 6435 [i]