MinT - Best Known (258−157, 258, s)-Nets in Base 2

Best Known (258−157, 258, s)-Nets in Base 2

(258−157, 258, 55)-Net over F₂ — Constructive and digital

Digital (101, 258, 55)-net over F₂, using

t-expansion [i] based on digital (100, 258, 55)-net over F₂, using
- net from sequence [i] based on digital (100, 54)-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 6 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]

(258−157, 258, 65)-Net over F₂ — Digital

Digital (101, 258, 65)-net over F₂, using

t-expansion [i] based on digital (95, 258, 65)-net over F₂, using
- net from sequence [i] based on digital (95, 64)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 95 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(258−157, 258, 194)-Net in Base 2 — Upper bound on s

There is no (101, 258, 195)-net in base 2, because

1 times m-reduction [i] would yield (101, 257, 195)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 301244 371731 179636 043030 480919 639663 150521 443300 925320 689026 482197 138884 607648 > 2²⁵⁷ [i]