MinT - Best Known (184−39, 184, s)-Nets in Base 3

Best Known (184−39, 184, s)-Nets in Base 3

(184−39, 184, 688)-Net over F₃ — Constructive and digital

Digital (145, 184, 688)-net over F₃, using

trace code for nets [i] based on digital (7, 46, 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]

(184−39, 184, 1644)-Net over F₃ — Digital

Digital (145, 184, 1644)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸⁴, 1644, F₃, 39) (dual of [1644, 1460, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁸⁴, 2203, F₃, 39) (dual of [2203, 2019, 40]-code), using
  - constructionÂ X applied to C([0,19]) âŠ‚ C([0,18]) [i] based on
    1. 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]
    2. linear OA(3¹⁶⁹, 2188, F₃, 37) (dual of [2188, 2019, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
    3. linear OA(3¹, 15, F₃, 1) (dual of [15, 14, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(184−39, 184, 156158)-Net in Base 3 — Upper bound on s

There is no (145, 184, 156159)-net in base 3, because

1 times m-reduction [i] would yield (145, 183, 156159)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2056 961265 279347 892811 014189 942889 864479 078297 988048 229515 710212 884975 335560 641355 868123 > 3¹⁸³ [i]