MinT - Best Known (63, 63+134, s)-Nets in Base 3

Best Known (63, 63+134, s)-Nets in Base 3

(63, 63+134, 48)-Net over F₃ — Constructive and digital

Digital (63, 197, 48)-net over F₃, using

t-expansion [i] based on digital (45, 197, 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+134, 64)-Net over F₃ — Digital

Digital (63, 197, 64)-net over F₃, using

t-expansion [i] based on digital (49, 197, 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+134, 198)-Net over F₃ — Upper bound on s (digital)

There is no digital (63, 197, 199)-net over F₃, because

5 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+134, 264)-Net in Base 3 — Upper bound on s

There is no (63, 197, 265)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10696 611469 616176 076636 299506 281424 626148 984020 473985 059120 854866 692963 903160 041265 699028 773499 > 3¹⁹⁷ [i]