Best Known (127−27, 127, s)-Nets in Base 3
(127−27, 127, 600)-Net over F3 — Constructive and digital
Digital (100, 127, 600)-net over F3, using
- 1 times m-reduction [i] based on digital (100, 128, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 32, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- trace code for nets [i] based on digital (4, 32, 150)-net over F81, using
(127−27, 127, 1269)-Net over F3 — Digital
Digital (100, 127, 1269)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3127, 1269, F3, 27) (dual of [1269, 1142, 28]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(3127, 2188, F3, 27) (dual of [2188, 2061, 28]-code), using
(127−27, 127, 119326)-Net in Base 3 — Upper bound on s
There is no (100, 127, 119327)-net in base 3, because
- 1 times m-reduction [i] would yield (100, 126, 119327)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 310099 806360 063617 189069 971424 012589 903269 611138 755786 515239 > 3126 [i]