MinT - Best Known (57, 57+135, s)-Nets in Base 3

Best Known (57, 57+135, s)-Nets in Base 3

(57, 57+135, 48)-Net over F₃ — Constructive and digital

Digital (57, 192, 48)-net over F₃, using

t-expansion [i] based on digital (45, 192, 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]

(57, 57+135, 64)-Net over F₃ — Digital

Digital (57, 192, 64)-net over F₃, using

t-expansion [i] based on digital (49, 192, 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]

(57, 57+135, 179)-Net over F₃ — Upper bound on s (digital)

There is no digital (57, 192, 180)-net over F₃, because

18 times m-reduction [i] would yield digital (57, 174, 180)-net over F₃, but
- extracting embedded orthogonal array [i] would yield 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]

(57, 57+135, 184)-Net in Base 3 — Upper bound on s

There is no (57, 192, 185)-net in base 3, because

12 times m-reduction [i] would yield (57, 180, 185)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁸⁰, 185, S₃, 123), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 6170 365191 715177 779482 467945 369860 501784 408913 594010 934644 126825 341528 124785 676079 587681 / 62 > 3¹⁸⁰ [i]