MinT - Best Known (137−37, 137, s)-Nets in Base 8

Best Known (137−37, 137, s)-Nets in Base 8

(137−37, 137, 1026)-Net over F₈ — Constructive and digital

Digital (100, 137, 1026)-net over F₈, using

7 times m-reduction [i] based on digital (100, 144, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 72, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(137−37, 137, 5595)-Net over F₈ — Digital

Digital (100, 137, 5595)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−37, 137, 7184191)-Net in Base 8 — Upper bound on s

There is no (100, 137, 7184192)-net in base 8, because

1 times m-reduction [i] would yield (100, 136, 7184192)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 661 056371 977730 966216 542262 378714 546598 732714 250659 483101 505664 836873 627722 682797 739710 800095 628156 614744 133929 338849 008165 > 8¹³⁶ [i]