Best Known (44, 44+7, s)-Nets in Base 5
(44, 44+7, 2796200)-Net over F5 — Constructive and digital
Digital (44, 51, 2796200)-net over F5, using
- net defined by OOA [i] based on linear OOA(551, 2796200, F5, 7, 7) (dual of [(2796200, 7), 19573349, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(551, 8388601, F5, 7) (dual of [8388601, 8388550, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(551, 8388601, F5, 7) (dual of [8388601, 8388550, 8]-code), using
(44, 44+7, 6360278)-Net over F5 — Digital
Digital (44, 51, 6360278)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(551, 6360278, F5, 7) (dual of [6360278, 6360227, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
(44, 44+7, large)-Net in Base 5 — Upper bound on s
There is no (44, 51, large)-net in base 5, because
- 5 times m-reduction [i] would yield (44, 46, large)-net in base 5, but