MinT - Best Known (239−168, 239, s)-Nets in Base 3

Best Known (239−168, 239, s)-Nets in Base 3

(239−168, 239, 48)-Net over F₃ — Constructive and digital

Digital (71, 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]

(239−168, 239, 84)-Net over F₃ — Digital

Digital (71, 239, 84)-net over F₃, using

net from sequence [i] based on digital (71, 83)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(239−168, 239, 222)-Net over F₃ — Upper bound on s (digital)

There is no digital (71, 239, 223)-net over F₃, because

24 times m-reduction [i] would yield digital (71, 215, 223)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²¹⁵, 223, F₃, 144) (dual of [223, 8, 145]-code), but
  - residual code [i] would yield linear OA(3⁷¹, 78, F₃, 48) (dual of [78, 7, 49]-code), but
    - residual code [i] would yield linear OA(3²³, 29, F₃, 16) (dual of [29, 6, 17]-code), but
      - 1 times truncation [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]

(239−168, 239, 226)-Net in Base 3 — Upper bound on s

There is no (71, 239, 227)-net in base 3, because

17 times m-reduction [i] would yield (71, 222, 227)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²²², 227, S₃, 151), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 675155 121849 745212 129793 196759 870681 725592 001960 937203 444022 441244 641816 765582 369161 757600 591162 976283 304329 / 76 > 3²²² [i]