MinT - Best Known (82, 82+59, s)-Nets in Base 2

Best Known (82, 82+59, s)-Nets in Base 2

(82, 82+59, 54)-Net over F₂ — Constructive and digital

Digital (82, 141, 54)-net over F₂, using

3 times m-reduction [i] based on digital (82, 144, 54)-net over F₂, using
- trace code for nets [i] based on digital (10, 72, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]

(82, 82+59, 63)-Net over F₂ — Digital

Digital (82, 141, 63)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+59, 290)-Net in Base 2 — Upper bound on s

There is no (82, 141, 291)-net in base 2, because

1 times m-reduction [i] would yield (82, 140, 291)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 419537 185541 540837 128849 314053 391685 603328 > 2¹⁴⁰ [i]