MinT - Best Known (53, 53+107, s)-Nets in Base 3

Best Known (53, 53+107, s)-Nets in Base 3

(53, 53+107, 48)-Net over F₃ — Constructive and digital

Digital (53, 160, 48)-net over F₃, using

t-expansion [i] based on digital (45, 160, 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]

(53, 53+107, 64)-Net over F₃ — Digital

Digital (53, 160, 64)-net over F₃, using

t-expansion [i] based on digital (49, 160, 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]

(53, 53+107, 174)-Net over F₃ — Upper bound on s (digital)

There is no digital (53, 160, 175)-net over F₃, because

2 times m-reduction [i] would yield digital (53, 158, 175)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁸, 175, F₃, 105) (dual of [175, 17, 106]-code), but
  - residual code [i] would yield linear OA(3⁵³, 69, F₃, 35) (dual of [69, 16, 36]-code), but
    - â€œLPâ€ bound on codes from Brouwerâ€™s database [i]

(53, 53+107, 229)-Net in Base 3 — Upper bound on s

There is no (53, 160, 230)-net in base 3, because

1 times m-reduction [i] would yield (53, 159, 230)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8327 500203 319018 210789 214412 802182 794847 546178 169654 193287 992769 181126 135621 > 3¹⁵⁹ [i]