MinT - Best Known (216−79, 216, s)-Nets in Base 2

Best Known (216−79, 216, s)-Nets in Base 2

(216−79, 216, 76)-Net over F₂ — Constructive and digital

Digital (137, 216, 76)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 78, 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, 138, 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]

(216−79, 216, 84)-Net in Base 2 — Constructive

(137, 216, 84)-net in base 2, using

4 times m-reduction [i] based on (137, 220, 84)-net in base 2, using
- trace code for nets [i] based on (27, 110, 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]

(216−79, 216, 121)-Net over F₂ — Digital

Digital (137, 216, 121)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−79, 216, 647)-Net in Base 2 — Upper bound on s

There is no (137, 216, 648)-net in base 2, because

1 times m-reduction [i] would yield (137, 215, 648)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 55399 578548 114811 499848 821486 710112 369765 972316 821023 118749 700308 > 2²¹⁵ [i]