MinT - Best Known (104, 128, s)-Nets in Base 3

Best Known (104, 128, s)-Nets in Base 3

(104, 128, 688)-Net over F₃ — Constructive and digital

Digital (104, 128, 688)-net over F₃, using

t-expansion [i] based on digital (103, 128, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 32, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(104, 128, 3144)-Net over F₃ — Digital

Digital (104, 128, 3144)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹²⁸, 3144, F₃, 2, 24) (dual of [(3144, 2), 6160, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹²⁸, 3280, F₃, 2, 24) (dual of [(3280, 2), 6432, 25]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹²⁸, 6560, F₃, 24) (dual of [6560, 6432, 25]-code), using
    - discarding factors / shortening the dual code based on linear OA(3¹²⁸, 6561, F₃, 24) (dual of [6561, 6433, 25]-code), using
      - 1 times truncation [i] based on linear OA(3¹²⁹, 6562, F₃, 25) (dual of [6562, 6433, 26]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(104, 128, 324795)-Net in Base 3 — Upper bound on s

There is no (104, 128, 324796)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 790590 790365 413734 682270 660760 682555 544049 587196 892115 353409 > 3¹²⁸ [i]