MinT - Best Known (248−87, 248, s)-Nets in Base 2

Best Known (248−87, 248, s)-Nets in Base 2

(248−87, 248, 112)-Net over F₂ — Constructive and digital

Digital (161, 248, 112)-net over F₂, using

8 times m-reduction [i] based on digital (161, 256, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 128, 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]

(248−87, 248, 149)-Net over F₂ — Digital

Digital (161, 248, 149)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−87, 248, 842)-Net in Base 2 — Upper bound on s

There is no (161, 248, 843)-net in base 2, because

1 times m-reduction [i] would yield (161, 247, 843)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 226 369768 259535 211091 584065 795051 778244 160111 291462 571374 012461 624964 675300 > 2²⁴⁷ [i]