MinT - Best Known (254−166, 254, s)-Nets in Base 2

Best Known (254−166, 254, s)-Nets in Base 2

(254−166, 254, 52)-Net over F₂ — Constructive and digital

Digital (88, 254, 52)-net over F₂, using

t-expansion [i] based on digital (85, 254, 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]

(254−166, 254, 57)-Net over F₂ — Digital

Digital (88, 254, 57)-net over F₂, using

t-expansion [i] based on digital (83, 254, 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]

(254−166, 254, 127)-Net in Base 2 — Upper bound on s

There is no (88, 254, 128)-net in base 2, because

7 times m-reduction [i] would yield (88, 247, 128)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁴⁷, 128, S₂, 2, 159), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 1243 860333 603982 568026 641440 523014 635142 548663 400435 746517 610765 710004 322304 / 5 > 2²⁴⁷ [i]