Best Known (36, 44, s)-Nets in Base 3
(36, 44, 1643)-Net over F3 — Constructive and digital
Digital (36, 44, 1643)-net over F3, using
- 31 times duplication [i] based on digital (35, 43, 1643)-net over F3, using
- net defined by OOA [i] based on linear OOA(343, 1643, F3, 8, 8) (dual of [(1643, 8), 13101, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(343, 6572, F3, 8) (dual of [6572, 6529, 9]-code), using
- 1 times code embedding in larger space [i] based on linear OA(342, 6571, F3, 8) (dual of [6571, 6529, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(341, 6561, F3, 8) (dual of [6561, 6520, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(333, 6561, F3, 7) (dual of [6561, 6528, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(39, 10, F3, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,3)), using
- dual of repetition code with length 10 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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 Ce(7) ⊂ Ce(6) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(342, 6571, F3, 8) (dual of [6571, 6529, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(343, 6572, F3, 8) (dual of [6572, 6529, 9]-code), using
- net defined by OOA [i] based on linear OOA(343, 1643, F3, 8, 8) (dual of [(1643, 8), 13101, 9]-NRT-code), using
(36, 44, 3927)-Net over F3 — Digital
Digital (36, 44, 3927)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(344, 3927, F3, 8) (dual of [3927, 3883, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(344, 6574, F3, 8) (dual of [6574, 6530, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(341, 6561, F3, 8) (dual of [6561, 6520, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(325, 6561, F3, 5) (dual of [6561, 6536, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(344, 6574, F3, 8) (dual of [6574, 6530, 9]-code), using
(36, 44, 196041)-Net in Base 3 — Upper bound on s
There is no (36, 44, 196042)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 984 773272 014872 845737 > 344 [i]