MinT - Best Known (68−42, 68, s)-Nets in Base 5

Best Known (68−42, 68, s)-Nets in Base 5

(68−42, 68, 51)-Net over F₅ — Constructive and digital

Digital (26, 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]

(68−42, 68, 55)-Net over F₅ — Digital

Digital (26, 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]

(68−42, 68, 382)-Net in Base 5 — Upper bound on s

There is no (26, 68, 383)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 341541 046244 424460 677287 538593 268739 302789 866925 > 5⁶⁸ [i]