MinT - Best Known (137−71, 137, s)-Nets in Base 5

Best Known (137−71, 137, s)-Nets in Base 5

(137−71, 137, 82)-Net over F₅ — Constructive and digital

Digital (66, 137, 82)-net over F₅, using

t-expansion [i] based on digital (48, 137, 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]

(137−71, 137, 120)-Net over F₅ — Digital

Digital (66, 137, 120)-net over F₅, using

t-expansion [i] based on digital (61, 137, 120)-net over F₅, using
- net from sequence [i] based on digital (61, 119)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 61 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(137−71, 137, 1782)-Net in Base 5 — Upper bound on s

There is no (66, 137, 1783)-net in base 5, because

1 times m-reduction [i] would yield (66, 136, 1783)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 116150 986174 454081 442093 303895 212372 672452 058990 769791 734575 825073 734806 952291 429051 916964 826325 > 5¹³⁶ [i]