MinT - Best Known (68, 212, s)-Nets in Base 3

Best Known (68, 212, s)-Nets in Base 3

(68, 212, 48)-Net over F₃ — Constructive and digital

Digital (68, 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]

(68, 212, 72)-Net over F₃ — Digital

Digital (68, 212, 72)-net over F₃, using

t-expansion [i] based on digital (67, 212, 72)-net over F₃, using
- net from sequence [i] based on digital (67, 71)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 67 and N(F) ≥ 72, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(68, 212, 213)-Net over F₃ — Upper bound on s (digital)

There is no digital (68, 212, 214)-net over F₃, because

6 times m-reduction [i] would yield digital (68, 206, 214)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁰⁶, 214, F₃, 138) (dual of [214, 8, 139]-code), but
  - residual code [i] would yield linear OA(3⁶⁸, 75, F₃, 46) (dual of [75, 7, 47]-code), but
    - 1 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]

(68, 212, 284)-Net in Base 3 — Upper bound on s

There is no (68, 212, 285)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 141894 580168 957079 464243 190960 177780 104568 229317 301824 028347 901819 852773 383894 308237 613326 008503 585377 > 3²¹² [i]