MinT - Best Known (209−81, 209, s)-Nets in Base 2

Best Known (209−81, 209, s)-Nets in Base 2

(209−81, 209, 68)-Net over F₂ — Constructive and digital

Digital (128, 209, 68)-net over F₂, using

5 times m-reduction [i] based on digital (128, 214, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 107, 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]

(209−81, 209, 103)-Net over F₂ — Digital

Digital (128, 209, 103)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−81, 209, 522)-Net in Base 2 — Upper bound on s

There is no (128, 209, 523)-net in base 2, because

1 times m-reduction [i] would yield (128, 208, 523)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 417 944785 633218 537468 242708 411173 207414 684354 325461 776076 649712 > 2²⁰⁸ [i]