Best Known (41, 41+9, s)-Nets in Base 3
(41, 41+9, 1644)-Net over F3 — Constructive and digital
Digital (41, 50, 1644)-net over F3, using
- net defined by OOA [i] based on linear OOA(350, 1644, F3, 9, 9) (dual of [(1644, 9), 14746, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(350, 6577, F3, 9) (dual of [6577, 6527, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(350, 6579, F3, 9) (dual of [6579, 6529, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(349, 6562, F3, 9) (dual of [6562, 6513, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(333, 6562, F3, 7) (dual of [6562, 6529, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(350, 6579, F3, 9) (dual of [6579, 6529, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(350, 6577, F3, 9) (dual of [6577, 6527, 10]-code), using
(41, 41+9, 3690)-Net over F3 — Digital
Digital (41, 50, 3690)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(350, 3690, F3, 9) (dual of [3690, 3640, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(350, 6579, F3, 9) (dual of [6579, 6529, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(349, 6562, F3, 9) (dual of [6562, 6513, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(333, 6562, F3, 7) (dual of [6562, 6529, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(350, 6579, F3, 9) (dual of [6579, 6529, 10]-code), using
(41, 41+9, 774027)-Net in Base 3 — Upper bound on s
There is no (41, 50, 774028)-net in base 3, because
- 1 times m-reduction [i] would yield (41, 49, 774028)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 239300 321078 825130 739329 > 349 [i]