MinT - Best Known (131, 158, s)-Nets in Base 3

Best Known (131, 158, s)-Nets in Base 3

(131, 158, 707)-Net over F₃ — Constructive and digital

Digital (131, 158, 707)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 22, 19)-net over F₃, using
  - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
      - an explicitly constructive algebraic function field [i]
2. digital (109, 136, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 34, 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]

(131, 158, 5022)-Net over F₃ — Digital

Digital (131, 158, 5022)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵⁸, 5022, F₃, 27) (dual of [5022, 4864, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁵⁸, 6576, F₃, 27) (dual of [6576, 6418, 28]-code), using
  - constructionÂ X applied to C([0,13]) âŠ‚ C([1,13]) [i] based on
    1. linear OA(3¹⁴⁵, 6562, F₃, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    2. linear OA(3¹⁴⁴, 6562, F₃, 13) (dual of [6562, 6418, 14]-code), using the narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    3. linear OA(3¹³, 14, F₃, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,3)), using
      - dual of repetition code with length 14 [i]

(131, 158, 1638813)-Net in Base 3 — Upper bound on s

There is no (131, 158, 1638814)-net in base 3, because

1 times m-reduction [i] would yield (131, 157, 1638814)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 809 167330 487009 146333 572617 998265 699566 214196 030895 124454 254258 183098 895685 > 3¹⁵⁷ [i]