MinT - Best Known (163, 163+81, s)-Nets in Base 2

Best Known (163, 163+81, s)-Nets in Base 2

(163, 163+81, 112)-Net over F₂ — Constructive and digital

Digital (163, 244, 112)-net over F₂, using

16 times m-reduction [i] based on digital (163, 260, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 130, 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]

(163, 163+81, 167)-Net over F₂ — Digital

Digital (163, 244, 167)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(163, 163+81, 1005)-Net in Base 2 — Upper bound on s

There is no (163, 244, 1006)-net in base 2, because

1 times m-reduction [i] would yield (163, 243, 1006)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 14 667616 528750 178797 909479 691811 028242 009717 159416 208572 719480 055043 187045 > 2²⁴³ [i]