MinT - Best Known (158−105, 158, s)-Nets in Base 3

Best Known (158−105, 158, s)-Nets in Base 3

(158−105, 158, 48)-Net over F₃ — Constructive and digital

Digital (53, 158, 48)-net over F₃, using

t-expansion [i] based on digital (45, 158, 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]

(158−105, 158, 64)-Net over F₃ — Digital

Digital (53, 158, 64)-net over F₃, using

t-expansion [i] based on digital (49, 158, 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]

(158−105, 158, 174)-Net over F₃ — Upper bound on s (digital)

There is no digital (53, 158, 175)-net over F₃, because

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]

(158−105, 158, 230)-Net in Base 3 — Upper bound on s

There is no (53, 158, 231)-net in base 3, because

1 times m-reduction [i] would yield (53, 157, 231)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 812 084991 583128 879698 351514 093005 597128 341281 954236 680572 474776 595152 834329 > 3¹⁵⁷ [i]