MinT - Best Known (164−109, 164, s)-Nets in Base 3

Best Known (164−109, 164, s)-Nets in Base 3

(164−109, 164, 48)-Net over F₃ — Constructive and digital

Digital (55, 164, 48)-net over F₃, using

t-expansion [i] based on digital (45, 164, 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]

(164−109, 164, 64)-Net over F₃ — Digital

Digital (55, 164, 64)-net over F₃, using

t-expansion [i] based on digital (49, 164, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(164−109, 164, 176)-Net over F₃ — Upper bound on s (digital)

There is no digital (55, 164, 177)-net over F₃, because

1 times m-reduction [i] would yield digital (55, 163, 177)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁶³, 177, F₃, 108) (dual of [177, 14, 109]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3¹⁶², 171, F₃, 108) (dual of [171, 9, 109]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(3¹⁶¹, 167, F₃, 108) (dual of [167, 6, 109]-code), but
        residual code [i] would yield 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]
        OA(3⁹, 171, S₃, 4), but
        discarding factors would yield OA(3⁹, 100, S₃, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 20001 > 3⁹ [i]
    2. OA(3¹⁴, 177, S₃, 6), but
      - discarding factors would yield OA(3¹⁴, 154, S₃, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 4 822665 > 3¹⁴ [i]

(164−109, 164, 239)-Net in Base 3 — Upper bound on s

There is no (55, 164, 240)-net in base 3, because

1 times m-reduction [i] would yield (55, 163, 240)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 694728 109216 096939 060819 018298 317696 285658 223923 116723 421865 834945 058154 313121 > 3¹⁶³ [i]