MinT - Best Known (63, 63+135, s)-Nets in Base 3

Best Known (63, 63+135, s)-Nets in Base 3

(63, 63+135, 48)-Net over F₃ — Constructive and digital

Digital (63, 198, 48)-net over F₃, using

t-expansion [i] based on digital (45, 198, 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]

(63, 63+135, 64)-Net over F₃ — Digital

Digital (63, 198, 64)-net over F₃, using

t-expansion [i] based on digital (49, 198, 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]

(63, 63+135, 198)-Net over F₃ — Upper bound on s (digital)

There is no digital (63, 198, 199)-net over F₃, because

6 times m-reduction [i] would yield digital (63, 192, 199)-net over F₃, but
- extracting embedded orthogonal array [i] would yield 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]

(63, 63+135, 202)-Net in Base 3 — Upper bound on s

There is no (63, 198, 203)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3¹⁹⁸, 203, S₃, 135), but
- the (dual) Plotkin bound shows that MÂ â‰¥ 2 390525 899882 872924 049031 898322 016641 463101 073880 550463 771174 655651 832418 111719 646949 462291 396009 / 68 > 3¹⁹⁸ [i]