MinT - Best Known (109−21, 109, s)-Nets in Base 3

Best Known (109−21, 109, s)-Nets in Base 3

(109−21, 109, 640)-Net over F₃ — Constructive and digital

Digital (88, 109, 640)-net over F₃, using

3¹ times duplication [i] based on digital (87, 108, 640)-net over F₃, using
- t-expansion [i] based on digital (86, 108, 640)-net over F₃, using
  - trace code for nets [i] based on digital (5, 27, 160)-net over F₈₁, using
    - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
        a function field by SÃ©mirat [i]

(109−21, 109, 2025)-Net over F₃ — Digital

Digital (88, 109, 2025)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁰⁹, 2025, F₃, 21) (dual of [2025, 1916, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁰⁹, 2199, F₃, 21) (dual of [2199, 2090, 22]-code), using
  - constructionÂ X applied to C([0,10]) âŠ‚ C([1,10]) [i] based on
    1. linear OA(3⁹⁹, 2188, F₃, 21) (dual of [2188, 2089, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
    2. linear OA(3⁹⁸, 2188, F₃, 10) (dual of [2188, 2090, 11]-code), using the narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(3¹⁰, 11, F₃, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,3)), using
      - dual of repetition code with length 11 [i]

(109−21, 109, 321990)-Net in Base 3 — Upper bound on s

There is no (88, 109, 321991)-net in base 3, because

1 times m-reduction [i] would yield (88, 108, 321991)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3381 445427 580502 398104 294038 891582 605827 856129 112317 > 3¹⁰⁸ [i]