MinT - Best Known (42, 42+56, s)-Nets in Base 5

Best Known (42, 42+56, s)-Nets in Base 5

(42, 42+56, 78)-Net over F₅ — Constructive and digital

Digital (42, 98, 78)-net over F₅, using

t-expansion [i] based on digital (38, 98, 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+56, 80)-Net over F₅ — Digital

Digital (42, 98, 80)-net over F₅, using

t-expansion [i] based on digital (41, 98, 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+56, 769)-Net in Base 5 — Upper bound on s

There is no (42, 98, 770)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 325 007776 709588 525104 426283 156130 796870 174635 035585 774564 168438 692545 > 5⁹⁸ [i]