MinT - Best Known (69, 212, s)-Nets in Base 3

Best Known (69, 212, s)-Nets in Base 3

(69, 212, 48)-Net over F₃ — Constructive and digital

Digital (69, 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]

(69, 212, 82)-Net over F₃ — Digital

Digital (69, 212, 82)-net over F₃, using

net from sequence [i] based on digital (69, 81)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 69 and N(F) ≥ 82, using
  - Drinfeld modules of rank 1 [i]

(69, 212, 217)-Net over F₃ — Upper bound on s (digital)

There is no digital (69, 212, 218)-net over F₃, because

2 times m-reduction [i] would yield digital (69, 210, 218)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²¹⁰, 218, F₃, 141) (dual of [218, 8, 142]-code), but
  - residual code [i] would yield linear OA(3⁶⁹, 76, F₃, 47) (dual of [76, 7, 48]-code), but
    - 2 times truncation [i] would yield linear OA(3⁶⁷, 74, F₃, 45) (dual of [74, 7, 46]-code), but
      - residual code [i] would yield linear OA(3²², 28, F₃, 15) (dual of [28, 6, 16]-code), but
        â€œHHMâ€ bound on codes from Brouwerâ€™s database [i]

(69, 212, 291)-Net in Base 3 — Upper bound on s

There is no (69, 212, 292)-net in base 3, because

1 times m-reduction [i] would yield (69, 211, 292)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 50734 215491 134208 514663 801204 684130 356557 021027 489253 571793 060483 551236 753099 081336 670654 458740 031345 > 3²¹¹ [i]