MinT - Best Known (110−33, 110, s)-Nets in Base 7

Best Known (110−33, 110, s)-Nets in Base 7

(110−33, 110, 688)-Net over F₇ — Constructive and digital

Digital (77, 110, 688)-net over F₇, using

t-expansion [i] based on digital (76, 110, 688)-net over F₇, using
- trace code for nets [i] based on digital (21, 55, 344)-net over F₄₉, using
  - net from sequence [i] based on digital (21, 343)-sequence over F₄₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 21 and N(F) ≥ 344, using
      - the Hermitian function field over F₄₉ [i]

(110−33, 110, 1729)-Net over F₇ — Digital

Digital (77, 110, 1729)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−33, 110, 648053)-Net in Base 7 — Upper bound on s

There is no (77, 110, 648054)-net in base 7, because

1 times m-reduction [i] would yield (77, 109, 648054)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 130 523452 056507 161659 967696 653787 075291 326945 148539 781454 301323 706312 454432 465334 616417 990945 > 7¹⁰⁹ [i]