MinT - Best Known (120, 120+59, s)-Nets in Base 2

Best Known (120, 120+59, s)-Nets in Base 2

(120, 120+59, 72)-Net over F₂ — Constructive and digital

Digital (120, 179, 72)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (37, 66, 30)-net over F₂, using
  - trace code for nets [i] based on digital (4, 33, 15)-net over F₄, using
    - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
        an explicitly constructive algebraic function field [i]
2. digital (54, 113, 42)-net over F₂, using
  - net from sequence [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]

(120, 120+59, 86)-Net in Base 2 — Constructive

(120, 179, 86)-net in base 2, using

1 times m-reduction [i] based on (120, 180, 86)-net in base 2, using
- trace code for nets [i] based on (30, 90, 43)-net in base 4, using
  - net from sequence [i] based on (30, 42)-sequence in base 4, using
    - base expansion [i] based on digital (60, 42)-sequence over F₂, using
      - t-expansion [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]

(120, 120+59, 131)-Net over F₂ — Digital

Digital (120, 179, 131)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(120, 120+59, 779)-Net in Base 2 — Upper bound on s

There is no (120, 179, 780)-net in base 2, because

1 times m-reduction [i] would yield (120, 178, 780)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 384370 836441 563138 715744 401391 763186 500217 594477 405248 > 2¹⁷⁸ [i]