Best Known (114, 114+30, s)-Nets in Base 3
(114, 114+30, 640)-Net over F3 — Constructive and digital
Digital (114, 144, 640)-net over F3, using
- t-expansion [i] based on digital (113, 144, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 36, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 36, 160)-net over F81, using
(114, 114+30, 1518)-Net over F3 — Digital
Digital (114, 144, 1518)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3144, 1518, F3, 30) (dual of [1518, 1374, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 2201, F3, 30) (dual of [2201, 2057, 31]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(3141, 2188, F3, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3127, 2188, F3, 27) (dual of [2188, 2061, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3144, 2201, F3, 30) (dual of [2201, 2057, 31]-code), using
(114, 114+30, 122193)-Net in Base 3 — Upper bound on s
There is no (114, 144, 122194)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 507 541418 056819 439769 720711 464322 305576 336084 601698 360422 976266 809945 > 3144 [i]