MinT - Best Known (49, 49+172, s)-Nets in Base 2

Best Known (49, 49+172, s)-Nets in Base 2

(49, 49+172, 35)-Net over F₂ — Constructive and digital

Digital (49, 221, 35)-net over F₂, using

t-expansion [i] based on digital (48, 221, 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]

(49, 49+172, 36)-Net over F₂ — Digital

Digital (49, 221, 36)-net over F₂, using

t-expansion [i] based on digital (47, 221, 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]

(49, 49+172, 64)-Net in Base 2 — Upper bound on s

There is no (49, 221, 65)-net in base 2, because

35 times m-reduction [i] would yield (49, 186, 65)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁸⁶, 65, S₂, 3, 137), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 25108 406941 546723 055343 157692 830665 664409 421777 856138 051584 / 207 > 2¹⁸⁶ [i]