MinT - Best Known (123−4, 123, s)-Nets in Base 3

Best Known (123−4, 123, s)-Nets in Base 3

(123−4, 123, large)-Net over F₃ — Constructive and digital

Digital (119, 123, large)-net over F₃, using

3³ times duplication [i] based on digital (116, 120, large)-net over F₃, using
- t-expansion [i] based on digital (112, 120, large)-net over F₃, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 4)-net over F₃, using
      - net from sequence [i] based on digital (0, 3)-sequence over F₃, using
        Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 0 and N(F) ≥ 4, using
        the rational function field F₃(x) [i]
        Niederreiter sequence [i]
    2. digital (108, 116, 8388600)-net over F₃, using
      - trace code for nets [i] based on digital (50, 58, 4194300)-net over F₉, using
        net defined by OOA [i] based on linear OOA(9⁵⁸, 4194300, F₉, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OOA(9⁵⁸, 8388601, F₉, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(9⁵⁸, 8388602, F₉, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
        trace code [i] based on linear OOA(81²⁹, 4194301, F₈₁, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81²⁹, 8388602, F₈₁, 8) (dual of [8388602, 8388573, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(81²⁹, large, F₈₁, 8) (dual of [large, large−29, 9]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(123−4, 123, large)-Net in Base 3 — Upper bound on s

There is no (119, 123, large)-net in base 3, because

2 times m-reduction [i] would yield (119, 121, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (123−4, 123, s)-Nets in Base 3

(123−4, 123, large)-Net over F3 — Constructive and digital

(123−4, 123, large)-Net in Base 3 — Upper bound on s

(123−4, 123, large)-Net over F₃ — Constructive and digital