MinT - Best Known (239−191, 239, s)-Nets in Base 2

Best Known (239−191, 239, s)-Nets in Base 2

(239−191, 239, 35)-Net over F₂ — Constructive and digital

Digital (48, 239, 35)-net over F₂, using

net from sequence [i] based on digital (48, 34)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(239−191, 239, 36)-Net over F₂ — Digital

Digital (48, 239, 36)-net over F₂, using

t-expansion [i] based on digital (47, 239, 36)-net over F₂, using
- net from sequence [i] based on digital (47, 35)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 47 and N(F) ≥ 36, using
    - Drinfeld modules of rank 1 [i]

(239−191, 239, 59)-Net in Base 2 — Upper bound on s

There is no (48, 239, 60)-net in base 2, because

9 times m-reduction [i] would yield (48, 230, 60)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²³⁰, 60, S₂, 4, 182), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 322656 641712 458857 062574 836561 450939 902089 344448 965571 453848 007462 617088 / 183 > 2²³⁰ [i]