Best Known (12, 12+7, s)-Nets in Base 3
(12, 12+7, 54)-Net over F3 — Constructive and digital
Digital (12, 19, 54)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 5, 20)-net over F3, using
- net defined by OOA [i] based on linear OOA(35, 20, F3, 3, 3) (dual of [(20, 3), 55, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(35, 20, F3, 2, 3) (dual of [(20, 2), 35, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(35, 20, F3, 3, 3) (dual of [(20, 3), 55, 4]-NRT-code), using
- digital (7, 14, 34)-net over F3, using
- digital (2, 5, 20)-net over F3, using
(12, 12+7, 64)-Net over F3 — Digital
Digital (12, 19, 64)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(319, 64, F3, 7) (dual of [64, 45, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(319, 85, F3, 7) (dual of [85, 66, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([1,3]) [i] based on
- linear OA(317, 82, F3, 7) (dual of [82, 65, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 82 | 38−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(316, 82, F3, 4) (dual of [82, 66, 5]-code), using the narrow-sense BCH-code C(I) with length 82 | 38−1, defining interval I = [1,3], and minimum distance d ≥ |{1,2,3}| + |{0,39,78,35}∖{39,35}| = 5 (general Roos-bound) [i]
- linear OA(32, 3, F3, 2) (dual of [3, 1, 3]-code or 3-arc in PG(1,3)), using
- dual of repetition code with length 3 [i]
- Reed–Solomon code RS(1,3) [i]
- construction X applied to C([0,3]) ⊂ C([1,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(319, 85, F3, 7) (dual of [85, 66, 8]-code), using
(12, 12+7, 659)-Net in Base 3 — Upper bound on s
There is no (12, 19, 660)-net in base 3, because
- 1 times m-reduction [i] would yield (12, 18, 660)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 387 693681 > 318 [i]