MinT - Best Known (107−65, 107, s)-Nets in Base 5

Best Known (107−65, 107, s)-Nets in Base 5

(107−65, 107, 78)-Net over F₅ — Constructive and digital

Digital (42, 107, 78)-net over F₅, using

t-expansion [i] based on digital (38, 107, 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]

(107−65, 107, 80)-Net over F₅ — Digital

Digital (42, 107, 80)-net over F₅, using

t-expansion [i] based on digital (41, 107, 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]

(107−65, 107, 637)-Net in Base 5 — Upper bound on s

There is no (42, 107, 638)-net in base 5, because

1 times m-reduction [i] would yield (42, 106, 638)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 124 890870 254006 332146 718426 343118 013992 011177 479396 132024 715994 186341 031425 > 5¹⁰⁶ [i]