MinT - Best Known (72, 72+159, s)-Nets in Base 3

Best Known (72, 72+159, s)-Nets in Base 3

(72, 72+159, 48)-Net over F₃ — Constructive and digital

Digital (72, 231, 48)-net over F₃, using

t-expansion [i] based on digital (45, 231, 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]

(72, 72+159, 84)-Net over F₃ — Digital

Digital (72, 231, 84)-net over F₃, using

t-expansion [i] based on digital (71, 231, 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]

(72, 72+159, 226)-Net over F₃ — Upper bound on s (digital)

There is no digital (72, 231, 227)-net over F₃, because

12 times m-reduction [i] would yield digital (72, 219, 227)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²¹⁹, 227, F₃, 147) (dual of [227, 8, 148]-code), but
  - residual code [i] would yield linear OA(3⁷², 79, F₃, 49) (dual of [79, 7, 50]-code), but
    - 1 times truncation [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]

(72, 72+159, 229)-Net in Base 3 — Upper bound on s

There is no (72, 231, 230)-net in base 3, because

6 times m-reduction [i] would yield (72, 225, 230)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²²⁵, 230, S₃, 153), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 18 229188 289943 120727 504416 312516 508406 590984 052945 304492 988605 913605 329052 670723 967367 455215 961400 359649 216883 / 77 > 3²²⁵ [i]