Best Known (37, 37+9, s)-Nets in Base 3
(37, 37+9, 551)-Net over F3 — Constructive and digital
Digital (37, 46, 551)-net over F3, using
- 31 times duplication [i] based on digital (36, 45, 551)-net over F3, using
- net defined by OOA [i] based on linear OOA(345, 551, F3, 9, 9) (dual of [(551, 9), 4914, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(345, 2205, F3, 9) (dual of [2205, 2160, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(344, 2204, F3, 9) (dual of [2204, 2160, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(343, 2188, F3, 9) (dual of [2188, 2145, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(329, 2188, F3, 7) (dual of [2188, 2159, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(315, 16, F3, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,3)), using
- dual of repetition code with length 16 [i]
- linear OA(31, 16, F3, 1) (dual of [16, 15, 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 X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(344, 2204, F3, 9) (dual of [2204, 2160, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(345, 2205, F3, 9) (dual of [2205, 2160, 10]-code), using
- net defined by OOA [i] based on linear OOA(345, 551, F3, 9, 9) (dual of [(551, 9), 4914, 10]-NRT-code), using
(37, 37+9, 1967)-Net over F3 — Digital
Digital (37, 46, 1967)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(346, 1967, F3, 9) (dual of [1967, 1921, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(346, 2206, F3, 9) (dual of [2206, 2160, 10]-code), using
- 2 times code embedding in larger space [i] based on linear OA(344, 2204, F3, 9) (dual of [2204, 2160, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(343, 2188, F3, 9) (dual of [2188, 2145, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(329, 2188, F3, 7) (dual of [2188, 2159, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(315, 16, F3, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,3)), using
- dual of repetition code with length 16 [i]
- linear OA(31, 16, F3, 1) (dual of [16, 15, 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 X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(344, 2204, F3, 9) (dual of [2204, 2160, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(346, 2206, F3, 9) (dual of [2206, 2160, 10]-code), using
(37, 37+9, 258006)-Net in Base 3 — Upper bound on s
There is no (37, 46, 258007)-net in base 3, because
- 1 times m-reduction [i] would yield (37, 45, 258007)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2954 324950 781742 534777 > 345 [i]