MinT - Best Known (42, 42+81, s)-Nets in Base 5

Best Known (42, 42+81, s)-Nets in Base 5

(42, 42+81, 78)-Net over F₅ — Constructive and digital

Digital (42, 123, 78)-net over F₅, using

t-expansion [i] based on digital (38, 123, 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]

(42, 42+81, 80)-Net over F₅ — Digital

Digital (42, 123, 80)-net over F₅, using

t-expansion [i] based on digital (41, 123, 80)-net over F₅, using
- net from sequence [i] based on digital (41, 79)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 41 and N(F) ≥ 80, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(42, 42+81, 505)-Net in Base 5 — Upper bound on s

There is no (42, 123, 506)-net in base 5, because

1 times m-reduction [i] would yield (42, 122, 506)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 20 081740 642552 137222 803488 552592 305570 400593 453901 945611 578755 493447 681349 519489 188225 > 5¹²² [i]