MinT - Best Known (184−95, 184, s)-Nets in Base 2

Best Known (184−95, 184, s)-Nets in Base 2

(184−95, 184, 52)-Net over F₂ — Constructive and digital

Digital (89, 184, 52)-net over F₂, using

t-expansion [i] based on digital (85, 184, 52)-net over F₂, using
- net from sequence [i] based on digital (85, 51)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 3 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(184−95, 184, 57)-Net over F₂ — Digital

Digital (89, 184, 57)-net over F₂, using

t-expansion [i] based on digital (83, 184, 57)-net over F₂, using
- net from sequence [i] based on digital (83, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
    - an algebraic function field by Auer [i]

(184−95, 184, 189)-Net over F₂ — Upper bound on s (digital)

There is no digital (89, 184, 190)-net over F₂, because

5 times m-reduction [i] would yield digital (89, 179, 190)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁷⁹, 190, F₂, 90) (dual of [190, 11, 91]-code), but
  - residual code [i] would yield linear OA(2⁸⁹, 99, F₂, 45) (dual of [99, 10, 46]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(184−95, 184, 210)-Net in Base 2 — Upper bound on s

There is no (89, 184, 211)-net in base 2, because

1 times m-reduction [i] would yield (89, 183, 211)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 14 029890 325356 732280 845101 748857 531595 955855 617066 477840 > 2¹⁸³ [i]