MinT - Best Known (58, 58+122, s)-Nets in Base 3

Best Known (58, 58+122, s)-Nets in Base 3

(58, 58+122, 48)-Net over F₃ — Constructive and digital

Digital (58, 180, 48)-net over F₃, using

t-expansion [i] based on digital (45, 180, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(58, 58+122, 64)-Net over F₃ — Digital

Digital (58, 180, 64)-net over F₃, using

t-expansion [i] based on digital (49, 180, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(58, 58+122, 183)-Net over F₃ — Upper bound on s (digital)

There is no digital (58, 180, 184)-net over F₃, because

5 times m-reduction [i] would yield digital (58, 175, 184)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁷⁵, 184, F₃, 117) (dual of [184, 9, 118]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3¹⁷⁴, 180, F₃, 117) (dual of [180, 6, 118]-code), but
      - residual code [i] would yield linear OA(3⁵⁷, 62, F₃, 39) (dual of [62, 5, 40]-code), but
        residual code [i] would yield linear OA(3¹⁸, 22, F₃, 13) (dual of [22, 4, 14]-code), but
        1 times truncation [i] would yield linear OA(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code), but
        â€œHNâ€ bound on codes from Brouwerâ€™s database [i]
    2. OA(3⁹, 184, S₃, 4), but
      - discarding factors would yield OA(3⁹, 100, S₃, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 20001 > 3⁹ [i]

(58, 58+122, 245)-Net in Base 3 — Upper bound on s

There is no (58, 180, 246)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 85 555341 306653 463487 248564 964393 185256 825016 245527 024986 928328 548986 747794 597868 621013 > 3¹⁸⁰ [i]