Best Known (244−46, 244, s)-Nets in Base 3
(244−46, 244, 896)-Net over F3 — Constructive and digital
Digital (198, 244, 896)-net over F3, using
- t-expansion [i] based on digital (196, 244, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 61, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 61, 224)-net over F81, using
(244−46, 244, 3681)-Net over F3 — Digital
Digital (198, 244, 3681)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3244, 3681, F3, 46) (dual of [3681, 3437, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3244, 6574, F3, 46) (dual of [6574, 6330, 47]-code), using
- construction X applied to Ce(45) ⊂ Ce(42) [i] based on
- linear OA(3241, 6561, F3, 46) (dual of [6561, 6320, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3225, 6561, F3, 43) (dual of [6561, 6336, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [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(45) ⊂ Ce(42) [i] based on
- discarding factors / shortening the dual code based on linear OA(3244, 6574, F3, 46) (dual of [6574, 6330, 47]-code), using
(244−46, 244, 543303)-Net in Base 3 — Upper bound on s
There is no (198, 244, 543304)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 261 574787 351912 497262 653891 004061 905062 665983 519143 560679 947604 486802 842980 087159 609557 291123 609787 190561 966908 789345 > 3244 [i]