MinT - Best Known (123−39, 123, s)-Nets in Base 2

Best Known (123−39, 123, s)-Nets in Base 2

(123−39, 123, 68)-Net over F₂ — Constructive and digital

Digital (84, 123, 68)-net over F₂, using

3 times m-reduction [i] based on digital (84, 126, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 63, 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]

(123−39, 123, 106)-Net over F₂ — Digital

Digital (84, 123, 106)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−39, 123, 651)-Net in Base 2 — Upper bound on s

There is no (84, 123, 652)-net in base 2, because

1 times m-reduction [i] would yield (84, 122, 652)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 5 325297 059837 672177 540961 294279 659864 > 2¹²² [i]