MinT - Best Known (168−114, 168, s)-Nets in Base 3

Best Known (168−114, 168, s)-Nets in Base 3

(168−114, 168, 48)-Net over F₃ — Constructive and digital

Digital (54, 168, 48)-net over F₃, using

t-expansion [i] based on digital (45, 168, 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]

(168−114, 168, 64)-Net over F₃ — Digital

Digital (54, 168, 64)-net over F₃, using

t-expansion [i] based on digital (49, 168, 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]

(168−114, 168, 170)-Net over F₃ — Upper bound on s (digital)

There is no digital (54, 168, 171)-net over F₃, because

6 times m-reduction [i] would yield digital (54, 162, 171)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁶², 171, F₃, 108) (dual of [171, 9, 109]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3¹⁶¹, 167, F₃, 108) (dual of [167, 6, 109]-code), but
      - residual code [i] would yield linear OA(3⁵³, 58, F₃, 36) (dual of [58, 5, 37]-code), but
        residual code [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, 171, 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]

(168−114, 168, 229)-Net in Base 3 — Upper bound on s

There is no (54, 168, 230)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 172 325644 107739 975301 878507 548236 621990 401120 429089 965538 112772 637471 833324 416733 > 3¹⁶⁸ [i]