MinT - Best Known (212−159, 212, s)-Nets in Base 3

Best Known (212−159, 212, s)-Nets in Base 3

(212−159, 212, 48)-Net over F₃ — Constructive and digital

Digital (53, 212, 48)-net over F₃, using

t-expansion [i] based on digital (45, 212, 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]

(212−159, 212, 64)-Net over F₃ — Digital

Digital (53, 212, 64)-net over F₃, using

t-expansion [i] based on digital (49, 212, 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]

(212−159, 212, 166)-Net over F₃ — Upper bound on s (digital)

There is no digital (53, 212, 167)-net over F₃, because

51 times m-reduction [i] would yield digital (53, 161, 167)-net over F₃, but
- extracting embedded orthogonal array [i] would yield 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]

(212−159, 212, 172)-Net in Base 3 — Upper bound on s

There is no (53, 212, 173)-net in base 3, because

44 times m-reduction [i] would yield (53, 168, 173)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁶⁸, 173, S₃, 115), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 11610 630703 530923 996233 764322 605633 554400 975674 804937 772291 047972 101377 433780 374641 / 58 > 3¹⁶⁸ [i]