MinT - Best Known (222−129, 222, s)-Nets in Base 2

Best Known (222−129, 222, s)-Nets in Base 2

(222−129, 222, 53)-Net over F₂ — Constructive and digital

Digital (93, 222, 53)-net over F₂, using

t-expansion [i] based on digital (90, 222, 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]

(222−129, 222, 60)-Net over F₂ — Digital

Digital (93, 222, 60)-net over F₂, using

t-expansion [i] based on digital (92, 222, 60)-net over F₂, using
- net from sequence [i] based on digital (92, 59)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 92 and N(F) ≥ 60, using
    - Drinfeld modules of rank 1 [i]

(222−129, 222, 187)-Net in Base 2 — Upper bound on s

There is no (93, 222, 188)-net in base 2, because

1 times m-reduction [i] would yield (93, 221, 188)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 593534 941946 587389 938837 638032 164105 651417 267040 998231 969609 750826 > 2²²¹ [i]