MinT - Best Known (158, 158+42, s)-Nets in Base 3

Best Known (158, 158+42, s)-Nets in Base 3

(158, 158+42, 688)-Net over F₃ — Constructive and digital

Digital (158, 200, 688)-net over F₃, using

t-expansion [i] based on digital (157, 200, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 50, 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]

(158, 158+42, 1827)-Net over F₃ — Digital

Digital (158, 200, 1827)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁰, 1827, F₃, 42) (dual of [1827, 1627, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁰⁰, 2201, F₃, 42) (dual of [2201, 2001, 43]-code), using
  - constructionÂ X applied to C([0,21]) âŠ‚ C([0,19]) [i] based on
    1. linear OA(3¹⁹⁷, 2188, F₃, 43) (dual of [2188, 1991, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
    2. linear OA(3¹⁸³, 2188, F₃, 39) (dual of [2188, 2005, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,19], and minimum distance dÂ â‰¥ |{−19,−18,…,19}|+1Â =Â 40 (BCH-bound) [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(158, 158+42, 151850)-Net in Base 3 — Upper bound on s

There is no (158, 200, 151851)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 265636 566493 952005 241302 110964 522521 194021 912751 540070 002893 231856 224316 335143 286705 081962 330863 > 3²⁰⁰ [i]