MinT - Best Known (116, 116+25, s)-Nets in Base 3

Best Known (116, 116+25, s)-Nets in Base 3

(116, 116+25, 695)-Net over F₃ — Constructive and digital

Digital (116, 141, 695)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. 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]

(116, 116+25, 3760)-Net over F₃ — Digital

Digital (116, 141, 3760)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁴¹, 3760, F₃, 25) (dual of [3760, 3619, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁴¹, 6575, F₃, 25) (dual of [6575, 6434, 26]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([1,12]) [i] based on
    1. 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]
    2. linear OA(3¹²⁸, 6562, F₃, 12) (dual of [6562, 6434, 13]-code), using the narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    3. linear OA(3¹², 13, F₃, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,3)), using
      - dual of repetition code with length 13 [i]

(116, 116+25, 974408)-Net in Base 3 — Upper bound on s

There is no (116, 141, 974409)-net in base 3, because

1 times m-reduction [i] would yield (116, 140, 974409)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 265849 065603 279790 526397 114217 642889 493993 786099 612550 531790 004657 > 3¹⁴⁰ [i]