Best Known (61−8, 61, s)-Nets in Base 5
(61−8, 61, 2097150)-Net over F5 — Constructive and digital
Digital (53, 61, 2097150)-net over F5, using
- net defined by OOA [i] based on linear OOA(561, 2097150, F5, 8, 8) (dual of [(2097150, 8), 16777139, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(561, 8388600, F5, 8) (dual of [8388600, 8388539, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(561, large, F5, 8) (dual of [large, large−61, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(561, large, F5, 8) (dual of [large, large−61, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(561, 8388600, F5, 8) (dual of [8388600, 8388539, 9]-code), using
(61−8, 61, 7309067)-Net over F5 — Digital
Digital (53, 61, 7309067)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(561, 7309067, F5, 8) (dual of [7309067, 7309006, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(561, large, F5, 8) (dual of [large, large−61, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(561, large, F5, 8) (dual of [large, large−61, 9]-code), using
(61−8, 61, large)-Net in Base 5 — Upper bound on s
There is no (53, 61, large)-net in base 5, because
- 6 times m-reduction [i] would yield (53, 55, large)-net in base 5, but