MinT - Best Known (222−83, 222, s)-Nets in Base 2

Best Known (222−83, 222, s)-Nets in Base 2

(222−83, 222, 76)-Net over F₂ — Constructive and digital

Digital (139, 222, 76)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 80, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (59, 142, 43)-net over F₂, using
  - net from sequence [i] based on digital (59, 42)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(222−83, 222, 84)-Net in Base 2 — Constructive

(139, 222, 84)-net in base 2, using

2 times m-reduction [i] based on (139, 224, 84)-net in base 2, using
- trace code for nets [i] based on (27, 112, 42)-net in base 4, using
  - net from sequence [i] based on (27, 41)-sequence in base 4, using
    - base expansion [i] based on digital (54, 41)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [i]

(222−83, 222, 118)-Net over F₂ — Digital

Digital (139, 222, 118)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−83, 222, 618)-Net in Base 2 — Upper bound on s

There is no (139, 222, 619)-net in base 2, because

1 times m-reduction [i] would yield (139, 221, 619)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 512271 966601 985720 828747 939593 247243 895080 082460 081470 894985 236780 > 2²²¹ [i]