MinT - Best Known (67, 239, s)-Nets in Base 3

Best Known (67, 239, s)-Nets in Base 3

(67, 239, 48)-Net over F₃ — Constructive and digital

Digital (67, 239, 48)-net over F₃, using

t-expansion [i] based on digital (45, 239, 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]

(67, 239, 72)-Net over F₃ — Digital

Digital (67, 239, 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]

(67, 239, 209)-Net over F₃ — Upper bound on s (digital)

There is no digital (67, 239, 210)-net over F₃, because

37 times m-reduction [i] would yield digital (67, 202, 210)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁰², 210, F₃, 135) (dual of [210, 8, 136]-code), but
  - residual code [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]

(67, 239, 214)-Net in Base 3 — Upper bound on s

There is no (67, 239, 215)-net in base 3, because

29 times m-reduction [i] would yield (67, 210, 215)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²¹⁰, 215, S₃, 143), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 141158 163862 183763 292171 284564 016760 661754 655311 572624 335224 092241 585052 457078 933432 718798 844642 935441 / 8 > 3²¹⁰ [i]