MinT - Best Known (58, 58+117, s)-Nets in Base 3

Best Known (58, 58+117, s)-Nets in Base 3

(58, 58+117, 48)-Net over F₃ — Constructive and digital

Digital (58, 175, 48)-net over F₃, using

t-expansion [i] based on digital (45, 175, 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+117, 64)-Net over F₃ — Digital

Digital (58, 175, 64)-net over F₃, using

t-expansion [i] based on digital (49, 175, 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+117, 183)-Net over F₃ — Upper bound on s (digital)

There is no digital (58, 175, 184)-net over F₃, because

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+117, 249)-Net in Base 3 — Upper bound on s

There is no (58, 175, 250)-net in base 3, because

1 times m-reduction [i] would yield (58, 174, 250)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 110546 142022 811316 756552 953792 311865 513316 348288 618817 367916 505233 557599 707756 306261 > 3¹⁷⁴ [i]