Best Known (48−7, 48, s)-Nets in Base 7
(48−7, 48, 1921599)-Net over F7 — Constructive and digital
Digital (41, 48, 1921599)-net over F7, using
- net defined by OOA [i] based on linear OOA(748, 1921599, F7, 7, 7) (dual of [(1921599, 7), 13451145, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(748, 5764798, F7, 7) (dual of [5764798, 5764750, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(748, 5764800, F7, 7) (dual of [5764800, 5764752, 8]-code), using
- the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(748, 5764800, F7, 7) (dual of [5764800, 5764752, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(748, 5764798, F7, 7) (dual of [5764798, 5764750, 8]-code), using
(48−7, 48, 5764800)-Net over F7 — Digital
Digital (41, 48, 5764800)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(748, 5764800, F7, 7) (dual of [5764800, 5764752, 8]-code), using
- the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
(48−7, 48, large)-Net in Base 7 — Upper bound on s
There is no (41, 48, large)-net in base 7, because
- 5 times m-reduction [i] would yield (41, 43, large)-net in base 7, but