MinT - Best Known (259−168, 259, s)-Nets in Base 2

Best Known (259−168, 259, s)-Nets in Base 2

(259−168, 259, 53)-Net over F₂ — Constructive and digital

Digital (91, 259, 53)-net over F₂, using

t-expansion [i] based on digital (90, 259, 53)-net over F₂, using
- net from sequence [i] based on digital (90, 52)-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 4 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]

(259−168, 259, 57)-Net over F₂ — Digital

Digital (91, 259, 57)-net over F₂, using

t-expansion [i] based on digital (83, 259, 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]

(259−168, 259, 131)-Net in Base 2 — Upper bound on s

There is no (91, 259, 132)-net in base 2, because

4 times m-reduction [i] would yield (91, 255, 132)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁵⁵, 132, S₂, 2, 164), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 926336 713898 529563 388567 880069 503262 826159 877325 124512 315660 672063 305037 119488 / 15 > 2²⁵⁵ [i]