MinT - Best Known (68−44, 68, s)-Nets in Base 5

Best Known (68−44, 68, s)-Nets in Base 5

(68−44, 68, 51)-Net over F₅ — Constructive and digital

Digital (24, 68, 51)-net over F₅, using

t-expansion [i] based on digital (22, 68, 51)-net over F₅, using
- net from sequence [i] based on digital (22, 50)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 22 and N(F) ≥ 51, using
    - an explicitly constructive algebraic function field [i]

(68−44, 68, 55)-Net over F₅ — Digital

Digital (24, 68, 55)-net over F₅, using

t-expansion [i] based on digital (23, 68, 55)-net over F₅, using
- net from sequence [i] based on digital (23, 54)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 23 and N(F) ≥ 55, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(68−44, 68, 311)-Net in Base 5 — Upper bound on s

There is no (24, 68, 312)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 343365 710223 706691 752791 767491 888871 638447 869825 > 5⁶⁸ [i]