Best Known (136, 136+36, s)-Nets in Base 3
(136, 136+36, 688)-Net over F3 — Constructive and digital
Digital (136, 172, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 43, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
(136, 136+36, 1667)-Net over F3 — Digital
Digital (136, 172, 1667)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3172, 1667, F3, 36) (dual of [1667, 1495, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3172, 2201, F3, 36) (dual of [2201, 2029, 37]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(3169, 2188, F3, 37) (dual of [2188, 2019, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3155, 2188, F3, 33) (dual of [2188, 2033, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (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([0,18]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3172, 2201, F3, 36) (dual of [2201, 2029, 37]-code), using
(136, 136+36, 136836)-Net in Base 3 — Upper bound on s
There is no (136, 172, 136837)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 11610 804952 072933 635281 408886 704487 342593 237228 921568 806742 118862 384335 170729 679761 > 3172 [i]