MinT - Best Known (68, 68+77, s)-Nets in Base 5

Best Known (68, 68+77, s)-Nets in Base 5

(68, 68+77, 82)-Net over F₅ — Constructive and digital

Digital (68, 145, 82)-net over F₅, using

t-expansion [i] based on digital (48, 145, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(68, 68+77, 132)-Net over F₅ — Digital

Digital (68, 145, 132)-net over F₅, using

t-expansion [i] based on digital (67, 145, 132)-net over F₅, using
- net from sequence [i] based on digital (67, 131)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 67 and N(F) ≥ 132, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(68, 68+77, 1645)-Net in Base 5 — Upper bound on s

There is no (68, 145, 1646)-net in base 5, because

1 times m-reduction [i] would yield (68, 144, 1646)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 45796 472072 097804 813270 116740 037019 815462 544483 818870 180951 010332 150482 900163 391106 047383 683464 383425 > 5¹⁴⁴ [i]