MinT - Best Known (78, 78+21, s)-Nets in Base 3

Best Known (78, 78+21, s)-Nets in Base 3

(78, 78+21, 464)-Net over F₃ — Constructive and digital

Digital (78, 99, 464)-net over F₃, using

t-expansion [i] based on digital (77, 99, 464)-net over F₃, using
- 1 times m-reduction [i] based on digital (77, 100, 464)-net over F₃, using
  - trace code for nets [i] based on digital (2, 25, 116)-net over F₈₁, using
    - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]

(78, 78+21, 1129)-Net over F₃ — Digital

Digital (78, 99, 1129)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁹⁹, 1129, F₃, 21) (dual of [1129, 1030, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁹⁹, 2188, F₃, 21) (dual of [2188, 2089, 22]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(78, 78+21, 107323)-Net in Base 3 — Upper bound on s

There is no (78, 99, 107324)-net in base 3, because

1 times m-reduction [i] would yield (78, 98, 107324)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 57265 073562 095634 002399 125674 369711 029956 656521 > 3⁹⁸ [i]