Best Known (56, 70, s)-Nets in Base 16
(56, 70, 149800)-Net over F16 — Constructive and digital
Digital (56, 70, 149800)-net over F16, using
- net defined by OOA [i] based on linear OOA(1670, 149800, F16, 14, 14) (dual of [(149800, 14), 2097130, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(1670, 1048600, F16, 14) (dual of [1048600, 1048530, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(1666, 1048576, F16, 14) (dual of [1048576, 1048510, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1646, 1048576, F16, 10) (dual of [1048576, 1048530, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(164, 24, F16, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,16)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- OA 7-folding and stacking [i] based on linear OA(1670, 1048600, F16, 14) (dual of [1048600, 1048530, 15]-code), using
(56, 70, 299593)-Net in Base 16 — Constructive
(56, 70, 299593)-net in base 16, using
- base change [i] based on digital (26, 40, 299593)-net over F128, using
- net defined by OOA [i] based on linear OOA(12840, 299593, F128, 14, 14) (dual of [(299593, 14), 4194262, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(12840, 2097151, F128, 14) (dual of [2097151, 2097111, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(12840, 2097151, F128, 14) (dual of [2097151, 2097111, 15]-code), using
- net defined by OOA [i] based on linear OOA(12840, 299593, F128, 14, 14) (dual of [(299593, 14), 4194262, 15]-NRT-code), using
(56, 70, 1150907)-Net over F16 — Digital
Digital (56, 70, 1150907)-net over F16, using
(56, 70, large)-Net in Base 16 — Upper bound on s
There is no (56, 70, large)-net in base 16, because
- 12 times m-reduction [i] would yield (56, 58, large)-net in base 16, but