Best Known (14, 14+15, s)-Nets in Base 64
(14, 14+15, 585)-Net over F64 — Constructive and digital
Digital (14, 29, 585)-net over F64, using
- net defined by OOA [i] based on linear OOA(6429, 585, F64, 15, 15) (dual of [(585, 15), 8746, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [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]
- OOA 7-folding and stacking with additional row [i] based on linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using
(14, 14+15, 1366)-Net over F64 — Digital
Digital (14, 29, 1366)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6429, 1366, F64, 3, 15) (dual of [(1366, 3), 4069, 16]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6429, 4098, F64, 15) (dual of [4098, 4069, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [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(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(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(14) ⊂ Ce(13) [i] based on
- OOA 3-folding [i] based on linear OA(6429, 4098, F64, 15) (dual of [4098, 4069, 16]-code), using
(14, 14+15, 900111)-Net in Base 64 — Upper bound on s
There is no (14, 29, 900112)-net in base 64, because
- 1 times m-reduction [i] would yield (14, 28, 900112)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 374 144769 775373 581393 196548 779078 865376 841344 148445 > 6428 [i]