MinT - Best Known (64, 64+129, s)-Nets in Base 3

Best Known (64, 64+129, s)-Nets in Base 3

(64, 64+129, 48)-Net over F₃ — Constructive and digital

Digital (64, 193, 48)-net over F₃, using

t-expansion [i] based on digital (45, 193, 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+129, 64)-Net over F₃ — Digital

Digital (64, 193, 64)-net over F₃, using

t-expansion [i] based on digital (49, 193, 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+129, 202)-Net over F₃ — Upper bound on s (digital)

There is no digital (64, 193, 203)-net over F₃, because

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+129, 274)-Net in Base 3 — Upper bound on s

There is no (64, 193, 275)-net in base 3, because

1 times m-reduction [i] would yield (64, 192, 275)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47 515392 343719 906135 764766 385485 468264 208426 625885 369140 622548 298175 604296 360940 083230 503809 > 3¹⁹² [i]