MinT - Best Known (68−45, 68, s)-Nets in Base 3

Best Known (68−45, 68, s)-Nets in Base 3

(68−45, 68, 32)-Net over F₃ — Constructive and digital

Digital (23, 68, 32)-net over F₃, using

t-expansion [i] based on digital (21, 68, 32)-net over F₃, using
- net from sequence [i] based on digital (21, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
    - an explicitly constructive algebraic function field [i]

(68−45, 68, 79)-Net over F₃ — Upper bound on s (digital)

There is no digital (23, 68, 80)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁶⁸, 80, F₃, 45) (dual of [80, 12, 46]-code), but
- constructionÂ Y1 [i] would yield
  1. 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(3¹², 80, S₃, 6), but
    - discarding factors would yield OA(3¹², 75, S₃, 6), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 551451 > 3¹² [i]

(68−45, 68, 81)-Net in Base 3 — Upper bound on s

There is no (23, 68, 82)-net in base 3, because

1 times m-reduction [i] would yield (23, 67, 82)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁶⁷, 82, S₃, 44), but
  - the linear programming bound shows that MÂ â‰¥ 11063 764601 718263 371177 932141 744468 313023 / 98 314060 > 3⁶⁷ [i]