Best Known (66, 66+12, s)-Nets in Base 5
(66, 66+12, 65109)-Net over F5 — Constructive and digital
Digital (66, 78, 65109)-net over F5, using
- net defined by OOA [i] based on linear OOA(578, 65109, F5, 12, 12) (dual of [(65109, 12), 781230, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(578, 390654, F5, 12) (dual of [390654, 390576, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(573, 390625, F5, 12) (dual of [390625, 390552, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(549, 390625, F5, 8) (dual of [390625, 390576, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(55, 29, F5, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- OA 6-folding and stacking [i] based on linear OA(578, 390654, F5, 12) (dual of [390654, 390576, 13]-code), using
(66, 66+12, 272882)-Net over F5 — Digital
Digital (66, 78, 272882)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(578, 272882, F5, 12) (dual of [272882, 272804, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(578, 390654, F5, 12) (dual of [390654, 390576, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(573, 390625, F5, 12) (dual of [390625, 390552, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(549, 390625, F5, 8) (dual of [390625, 390576, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(55, 29, F5, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(578, 390654, F5, 12) (dual of [390654, 390576, 13]-code), using
(66, 66+12, large)-Net in Base 5 — Upper bound on s
There is no (66, 78, large)-net in base 5, because
- 10 times m-reduction [i] would yield (66, 68, large)-net in base 5, but