Best Known (51−9, 51, s)-Nets in Base 3
(51−9, 51, 1645)-Net over F3 — Constructive and digital
Digital (42, 51, 1645)-net over F3, using
- net defined by OOA [i] based on linear OOA(351, 1645, F3, 9, 9) (dual of [(1645, 9), 14754, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(351, 6581, F3, 9) (dual of [6581, 6530, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(350, 6580, F3, 9) (dual of [6580, 6530, 10]-code), using
- construction X4 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(317, 18, F3, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,3)), using
- dual of repetition code with length 18 [i]
- linear OA(31, 18, F3, 1) (dual of [18, 17, 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(350, 6580, F3, 9) (dual of [6580, 6530, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(351, 6581, F3, 9) (dual of [6581, 6530, 10]-code), using
(51−9, 51, 4318)-Net over F3 — Digital
Digital (42, 51, 4318)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(351, 4318, F3, 9) (dual of [4318, 4267, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(351, 6581, F3, 9) (dual of [6581, 6530, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(350, 6580, F3, 9) (dual of [6580, 6530, 10]-code), using
- construction X4 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(317, 18, F3, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,3)), using
- dual of repetition code with length 18 [i]
- linear OA(31, 18, F3, 1) (dual of [18, 17, 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(350, 6580, F3, 9) (dual of [6580, 6530, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(351, 6581, F3, 9) (dual of [6581, 6530, 10]-code), using
(51−9, 51, 1018678)-Net in Base 3 — Upper bound on s
There is no (42, 51, 1018679)-net in base 3, because
- 1 times m-reduction [i] would yield (42, 50, 1018679)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 717900 280058 681329 201273 > 350 [i]