MinT - Best Known (68−50, 68, s)-Nets in Base 3

Best Known (68−50, 68, s)-Nets in Base 3

(68−50, 68, 28)-Net over F₃ — Constructive and digital

Digital (18, 68, 28)-net over F₃, using

t-expansion [i] based on digital (15, 68, 28)-net over F₃, using
- net from sequence [i] based on digital (15, 27)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
    - an explicitly constructive algebraic function field [i]

(68−50, 68, 61)-Net over F₃ — Upper bound on s (digital)

There is no digital (18, 68, 62)-net over F₃, because

14 times m-reduction [i] would yield digital (18, 54, 62)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁵⁴, 62, F₃, 36) (dual of [62, 8, 37]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3⁵³, 58, F₃, 36) (dual of [58, 5, 37]-code), but
      - residual code [i] would yield linear OA(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code), but
        â€œHNâ€ bound on codes from Brouwerâ€™s database [i]
    2. OA(3⁸, 62, S₃, 4), but
      - discarding factors would yield OA(3⁸, 58, S₃, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 6729 > 3⁸ [i]

(68−50, 68, 63)-Net in Base 3 — Upper bound on s

There is no (18, 68, 64)-net in base 3, because

11 times m-reduction [i] would yield (18, 57, 64)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁵⁷, 64, S₃, 39), but
  - the linear programming bound shows that MÂ â‰¥ 381520 424476 945831 628649 898809 / 235 > 3⁵⁷ [i]