MinT - Best Known (68, 68+137, s)-Nets in Base 3

Best Known (68, 68+137, s)-Nets in Base 3

(68, 68+137, 48)-Net over F₃ — Constructive and digital

Digital (68, 205, 48)-net over F₃, using

t-expansion [i] based on digital (45, 205, 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, 68+137, 72)-Net over F₃ — Digital

Digital (68, 205, 72)-net over F₃, using

t-expansion [i] based on digital (67, 205, 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, 68+137, 215)-Net over F₃ — Upper bound on s (digital)

There is no digital (68, 205, 216)-net over F₃, because

2 times m-reduction [i] would yield digital (68, 203, 216)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁰³, 216, F₃, 135) (dual of [216, 13, 136]-code), but
  - constructionÂ Y1 [i] would yield
    1. 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]
    2. OA(3¹³, 216, S₃, 6), but
      - discarding factors would yield OA(3¹³, 107, S₃, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1 610779 > 3¹³ [i]

(68, 68+137, 290)-Net in Base 3 — Upper bound on s

There is no (68, 205, 291)-net in base 3, because

1 times m-reduction [i] would yield (68, 204, 291)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 23 824984 473606 983640 329466 377648 844929 779949 730778 322913 853376 411662 002441 358642 313443 142764 174425 > 3²⁰⁴ [i]