MinT - Best Known (64, 64+178, s)-Nets in Base 3

Best Known (64, 64+178, s)-Nets in Base 3

(64, 64+178, 48)-Net over F₃ — Constructive and digital

Digital (64, 242, 48)-net over F₃, using

t-expansion [i] based on digital (45, 242, 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]

(64, 64+178, 64)-Net over F₃ — Digital

Digital (64, 242, 64)-net over F₃, using

t-expansion [i] based on digital (49, 242, 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]

(64, 64+178, 202)-Net over F₃ — Upper bound on s (digital)

There is no digital (64, 242, 203)-net over F₃, because

49 times m-reduction [i] would yield digital (64, 193, 203)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁹³, 203, F₃, 129) (dual of [203, 10, 130]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3¹⁹², 199, F₃, 129) (dual of [199, 7, 130]-code), but
      - residual code [i] would yield linear OA(3⁶³, 69, F₃, 43) (dual of [69, 6, 44]-code), but
        â€œMaâ€ bound on codes from Brouwerâ€™s database [i]
    2. OA(3¹⁰, 203, S₃, 4), but
      - discarding factors would yield OA(3¹⁰, 172, S₃, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 59169 > 3¹⁰ [i]

(64, 64+178, 205)-Net in Base 3 — Upper bound on s

There is no (64, 242, 206)-net in base 3, because

41 times m-reduction [i] would yield (64, 201, 206)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²⁰¹, 206, S₃, 137), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 21 514733 098945 856316 441287 084898 149773 167909 664924 954173 940571 900866 491763 005476 822545 160622 564081 / 23 > 3²⁰¹ [i]