MinT - Best Known (63, 63+129, s)-Nets in Base 3

Best Known (63, 63+129, s)-Nets in Base 3

(63, 63+129, 48)-Net over F₃ — Constructive and digital

Digital (63, 192, 48)-net over F₃, using

t-expansion [i] based on digital (45, 192, 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+129, 64)-Net over F₃ — Digital

Digital (63, 192, 64)-net over F₃, using

t-expansion [i] based on digital (49, 192, 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+129, 198)-Net over F₃ — Upper bound on s (digital)

There is no digital (63, 192, 199)-net over F₃, because

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+129, 268)-Net in Base 3 — Upper bound on s

There is no (63, 192, 269)-net in base 3, because

1 times m-reduction [i] would yield (63, 191, 269)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 14 667483 760095 766439 388037 267493 290534 199047 189870 901161 345683 026665 086892 195396 960298 870529 > 3¹⁹¹ [i]