Best Known (44−9, 44, s)-Nets in Base 3
(44−9, 44, 550)-Net over F3 — Constructive and digital
Digital (35, 44, 550)-net over F3, using
- net defined by OOA [i] based on linear OOA(344, 550, F3, 9, 9) (dual of [(550, 9), 4906, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(344, 2201, F3, 9) (dual of [2201, 2157, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(344, 2203, F3, 9) (dual of [2203, 2159, 10]-code), using
- construction X 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(31, 15, F3, 1) (dual of [15, 14, 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(344, 2203, F3, 9) (dual of [2203, 2159, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(344, 2201, F3, 9) (dual of [2201, 2157, 10]-code), using
(44−9, 44, 1435)-Net over F3 — Digital
Digital (35, 44, 1435)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(344, 1435, F3, 9) (dual of [1435, 1391, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(344, 2203, F3, 9) (dual of [2203, 2159, 10]-code), using
- construction X 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(31, 15, F3, 1) (dual of [15, 14, 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(344, 2203, F3, 9) (dual of [2203, 2159, 10]-code), using
(44−9, 44, 148958)-Net in Base 3 — Upper bound on s
There is no (35, 44, 148959)-net in base 3, because
- 1 times m-reduction [i] would yield (35, 43, 148959)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 328 258930 323081 282553 > 343 [i]