MinT - Best Known (181−121, 181, s)-Nets in Base 3

Best Known (181−121, 181, s)-Nets in Base 3

(181−121, 181, 48)-Net over F₃ — Constructive and digital

Digital (60, 181, 48)-net over F₃, using

t-expansion [i] based on digital (45, 181, 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]

(181−121, 181, 64)-Net over F₃ — Digital

Digital (60, 181, 64)-net over F₃, using

t-expansion [i] based on digital (49, 181, 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]

(181−121, 181, 191)-Net over F₃ — Upper bound on s (digital)

There is no digital (60, 181, 192)-net over F₃, because

1 times m-reduction [i] would yield digital (60, 180, 192)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁰, 192, F₃, 120) (dual of [192, 12, 121]-code), but
  - residual code [i] would yield linear OA(3⁶⁰, 71, F₃, 40) (dual of [71, 11, 41]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(181−121, 181, 257)-Net in Base 3 — Upper bound on s

There is no (60, 181, 258)-net in base 3, because

1 times m-reduction [i] would yield (60, 180, 258)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 78 167523 651657 376410 260929 586918 864081 268942 768349 527806 167604 299951 905493 352322 607449 > 3¹⁸⁰ [i]