MinT - Best Known (127, 127+63, s)-Nets in Base 2

Best Known (127, 127+63, s)-Nets in Base 2

(127, 127+63, 76)-Net over F₂ — Constructive and digital

Digital (127, 190, 76)-net over F₂, using

2¹ times duplication [i] based on digital (126, 189, 76)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (41, 72, 34)-net over F₂, using
    - trace code for nets [i] based on digital (5, 36, 17)-net over F₄, using
      - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 117, 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]

(127, 127+63, 86)-Net in Base 2 — Constructive

(127, 190, 86)-net in base 2, using

4 times m-reduction [i] based on (127, 194, 86)-net in base 2, using
- trace code for nets [i] based on (30, 97, 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]

(127, 127+63, 135)-Net over F₂ — Digital

Digital (127, 190, 135)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+63, 804)-Net in Base 2 — Upper bound on s

There is no (127, 190, 805)-net in base 2, because

1 times m-reduction [i] would yield (127, 189, 805)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 785 755618 402494 509162 350969 294056 514540 177595 500169 824608 > 2¹⁸⁹ [i]