MinT - Best Known (105−59, 105, s)-Nets in Base 5

Best Known (105−59, 105, s)-Nets in Base 5

(105−59, 105, 78)-Net over F₅ — Constructive and digital

Digital (46, 105, 78)-net over F₅, using

t-expansion [i] based on digital (38, 105, 78)-net over F₅, using
- net from sequence [i] based on digital (38, 77)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 38 and N(F) ≥ 78, using
    - an explicitly constructive algebraic function field [i]

(105−59, 105, 88)-Net over F₅ — Digital

Digital (46, 105, 88)-net over F₅, using

t-expansion [i] based on digital (45, 105, 88)-net over F₅, using
- net from sequence [i] based on digital (45, 87)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 45 and N(F) ≥ 88, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(105−59, 105, 915)-Net in Base 5 — Upper bound on s

There is no (46, 105, 916)-net in base 5, because

1 times m-reduction [i] would yield (46, 104, 916)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4 951230 303695 447687 095473 648374 312098 580166 218848 324495 089199 412274 495185 > 5¹⁰⁴ [i]