Best Known (50, 50+7, s)-Nets in Base 3
(50, 50+7, 1594323)-Net over F3 — Constructive and digital
Digital (50, 57, 1594323)-net over F3, using
- net defined by OOA [i] based on linear OOA(357, 1594323, F3, 7, 7) (dual of [(1594323, 7), 11160204, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(357, 4782970, F3, 7) (dual of [4782970, 4782913, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(357, 4782970, F3, 7) (dual of [4782970, 4782913, 8]-code), using
(50, 50+7, 2391485)-Net over F3 — Digital
Digital (50, 57, 2391485)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(357, 2391485, F3, 2, 7) (dual of [(2391485, 2), 4782913, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(357, 4782970, F3, 7) (dual of [4782970, 4782913, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 2-folding [i] based on linear OA(357, 4782970, F3, 7) (dual of [4782970, 4782913, 8]-code), using
(50, 50+7, large)-Net in Base 3 — Upper bound on s
There is no (50, 57, large)-net in base 3, because
- 5 times m-reduction [i] would yield (50, 52, large)-net in base 3, but