MinT - Best Known (107−29, 107, s)-Nets in Base 7

Best Known (107−29, 107, s)-Nets in Base 7

(107−29, 107, 688)-Net over F₇ — Constructive and digital

Digital (78, 107, 688)-net over F₇, using

t-expansion [i] based on digital (76, 107, 688)-net over F₇, using
- 3 times m-reduction [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]

(107−29, 107, 3208)-Net over F₇ — Digital

Digital (78, 107, 3208)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(107−29, 107, 2522925)-Net in Base 7 — Upper bound on s

There is no (78, 107, 2522926)-net in base 7, because

1 times m-reduction [i] would yield (78, 106, 2522926)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 380534 130141 451400 029463 175406 885026 515062 821599 651475 207236 549904 185851 809663 297165 984709 > 7¹⁰⁶ [i]