MinT - Best Known (104, 104+51, s)-Nets in Base 2

Best Known (104, 104+51, s)-Nets in Base 2

(104, 104+51, 68)-Net over F₂ — Constructive and digital

Digital (104, 155, 68)-net over F₂, using

11 times m-reduction [i] based on digital (104, 166, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 83, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(104, 104+51, 72)-Net in Base 2 — Constructive

(104, 155, 72)-net in base 2, using

1 times m-reduction [i] based on (104, 156, 72)-net in base 2, using
- trace code for nets [i] based on (26, 78, 36)-net in base 4, using
  - net from sequence [i] based on (26, 35)-sequence in base 4, using
    - base expansion [i] based on digital (52, 35)-sequence over F₂, using
      - t-expansion [i] based on digital (51, 35)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(104, 104+51, 116)-Net over F₂ — Digital

Digital (104, 155, 116)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(104, 104+51, 691)-Net in Base 2 — Upper bound on s

There is no (104, 155, 692)-net in base 2, because

1 times m-reduction [i] would yield (104, 154, 692)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 23179 320114 486211 482001 012872 268112 158463 763564 > 2¹⁵⁴ [i]