Best Known (144, 144+30, s)-Nets in Base 3
(144, 144+30, 896)-Net over F3 — Constructive and digital
Digital (144, 174, 896)-net over F3, using
- 32 times duplication [i] based on digital (142, 172, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 43, 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, 43, 224)-net over F81, using
(144, 144+30, 4984)-Net over F3 — Digital
Digital (144, 174, 4984)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3174, 4984, F3, 30) (dual of [4984, 4810, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3174, 6607, F3, 30) (dual of [6607, 6433, 31]-code), using
- 5 times code embedding in larger space [i] based on linear OA(3169, 6602, F3, 30) (dual of [6602, 6433, 31]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3161, 6562, F3, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3129, 6562, F3, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(38, 40, F3, 4) (dual of [40, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- 5 times code embedding in larger space [i] based on linear OA(3169, 6602, F3, 30) (dual of [6602, 6433, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3174, 6607, F3, 30) (dual of [6607, 6433, 31]-code), using
(144, 144+30, 1099860)-Net in Base 3 — Upper bound on s
There is no (144, 174, 1099861)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 104496 861151 028039 822755 591204 683904 989150 876106 532770 826186 455385 509896 686295 426491 > 3174 [i]