MinT - Best Known (88−53, 88, s)-Nets in Base 5

Best Known (88−53, 88, s)-Nets in Base 5

(88−53, 88, 72)-Net over F₅ — Constructive and digital

Digital (35, 88, 72)-net over F₅, using

t-expansion [i] based on digital (31, 88, 72)-net over F₅, using
- net from sequence [i] based on digital (31, 71)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
    - an explicitly constructive algebraic function field [i]

(88−53, 88, 76)-Net over F₅ — Digital

Digital (35, 88, 76)-net over F₅, using

t-expansion [i] based on digital (34, 88, 76)-net over F₅, using
- net from sequence [i] based on digital (34, 75)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 34 and N(F) ≥ 76, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(88−53, 88, 556)-Net in Base 5 — Upper bound on s

There is no (35, 88, 557)-net in base 5, because

1 times m-reduction [i] would yield (35, 87, 557)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 6 524976 912832 467855 502175 746964 055857 342392 227223 128888 250297 > 5⁸⁷ [i]