MinT - Best Known (157, 157+43, s)-Nets in Base 3

Best Known (157, 157+43, s)-Nets in Base 3

(157, 157+43, 688)-Net over F₃ — Constructive and digital

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]

(157, 157+43, 1632)-Net over F₃ — Digital

Digital (157, 200, 1632)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁰, 1632, F₃, 43) (dual of [1632, 1432, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁰⁰, 2192, F₃, 43) (dual of [2192, 1992, 44]-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, 4, F₃, 3) (dual of [4, 1, 4]-code or 4-arc in PG(2,3) or 4-cap in PG(2,3)), using
      - dual of repetition code with length 4 [i]
      - oval in PG(2,Â 3) [i]

(157, 157+43, 144109)-Net in Base 3 — Upper bound on s

There is no (157, 200, 144110)-net in base 3, because

1 times m-reduction [i] would yield (157, 199, 144110)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 88544 110227 412340 531021 526860 132468 312520 265543 151176 197462 627142 021001 052972 424816 944586 030677 > 3¹⁹⁹ [i]