MinT - Best Known (245−83, 245, s)-Nets in Base 2

Best Known (245−83, 245, s)-Nets in Base 2

(245−83, 245, 112)-Net over F₂ — Constructive and digital

Digital (162, 245, 112)-net over F₂, using

13 times m-reduction [i] based on digital (162, 258, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 129, 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]

(245−83, 245, 160)-Net over F₂ — Digital

Digital (162, 245, 160)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−83, 245, 939)-Net in Base 2 — Upper bound on s

There is no (162, 245, 940)-net in base 2, because

1 times m-reduction [i] would yield (162, 244, 940)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 29 105760 082069 667654 856200 022226 997380 854207 918451 809416 891721 048239 475237 > 2²⁴⁴ [i]