Best Known (28−14, 28, s)-Nets in Base 64
(28−14, 28, 585)-Net over F64 — Constructive and digital
Digital (14, 28, 585)-net over F64, using
- 1 times m-reduction [i] based on 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
- net defined by OOA [i] based on linear OOA(6429, 585, F64, 15, 15) (dual of [(585, 15), 8746, 16]-NRT-code), using
(28−14, 28, 1444)-Net over F64 — Digital
Digital (14, 28, 1444)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6428, 1444, F64, 2, 14) (dual of [(1444, 2), 2860, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6428, 2050, F64, 2, 14) (dual of [(2050, 2), 4072, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6428, 4100, F64, 14) (dual of [4100, 4072, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6428, 4101, F64, 14) (dual of [4101, 4073, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [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(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(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(6428, 4101, F64, 14) (dual of [4101, 4073, 15]-code), using
- OOA 2-folding [i] based on linear OA(6428, 4100, F64, 14) (dual of [4100, 4072, 15]-code), using
- discarding factors / shortening the dual code based on linear OOA(6428, 2050, F64, 2, 14) (dual of [(2050, 2), 4072, 15]-NRT-code), using
(28−14, 28, 900111)-Net in Base 64 — Upper bound on s
There is no (14, 28, 900112)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 374 144769 775373 581393 196548 779078 865376 841344 148445 > 6428 [i]