MinT - Best Known (159−53, 159, s)-Nets in Base 2

Best Known (159−53, 159, s)-Nets in Base 2

(159−53, 159, 68)-Net over F₂ — Constructive and digital

Digital (106, 159, 68)-net over F₂, using

11 times m-reduction [i] based on digital (106, 170, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 85, 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]

(159−53, 159, 72)-Net in Base 2 — Constructive

(106, 159, 72)-net in base 2, using

1 times m-reduction [i] based on (106, 160, 72)-net in base 2, using
- trace code for nets [i] based on (26, 80, 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]

(159−53, 159, 115)-Net over F₂ — Digital

Digital (106, 159, 115)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(159−53, 159, 674)-Net in Base 2 — Upper bound on s

There is no (106, 159, 675)-net in base 2, because

1 times m-reduction [i] would yield (106, 158, 675)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 369841 774163 528187 843050 534811 654795 735380 286028 > 2¹⁵⁸ [i]