Best Known (68−12, 68, s)-Nets in Base 5
(68−12, 68, 13025)-Net over F5 — Constructive and digital
Digital (56, 68, 13025)-net over F5, using
- net defined by OOA [i] based on linear OOA(568, 13025, F5, 12, 12) (dual of [(13025, 12), 156232, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(568, 78150, F5, 12) (dual of [78150, 78082, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(543, 78125, F5, 8) (dual of [78125, 78082, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(54, 25, F5, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,5)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- OA 6-folding and stacking [i] based on linear OA(568, 78150, F5, 12) (dual of [78150, 78082, 13]-code), using
(68−12, 68, 54571)-Net over F5 — Digital
Digital (56, 68, 54571)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(568, 54571, F5, 12) (dual of [54571, 54503, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(568, 78150, F5, 12) (dual of [78150, 78082, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(543, 78125, F5, 8) (dual of [78125, 78082, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(54, 25, F5, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,5)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(568, 78150, F5, 12) (dual of [78150, 78082, 13]-code), using
(68−12, 68, large)-Net in Base 5 — Upper bound on s
There is no (56, 68, large)-net in base 5, because
- 10 times m-reduction [i] would yield (56, 58, large)-net in base 5, but