MinT - Best Known (222−77, 222, s)-Nets in Base 2

Best Known (222−77, 222, s)-Nets in Base 2

(222−77, 222, 112)-Net over F₂ — Constructive and digital

Digital (145, 222, 112)-net over F₂, using

2 times m-reduction [i] based on digital (145, 224, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 112, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(222−77, 222, 139)-Net over F₂ — Digital

Digital (145, 222, 139)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−77, 222, 791)-Net in Base 2 — Upper bound on s

There is no (145, 222, 792)-net in base 2, because

1 times m-reduction [i] would yield (145, 221, 792)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 450090 366026 118646 521923 751760 833395 444885 495819 770822 318689 055400 > 2²²¹ [i]