MinT - Best Known (51, 51+57, s)-Nets in Base 5

Best Known (51, 51+57, s)-Nets in Base 5

(51, 51+57, 82)-Net over F₅ — Constructive and digital

Digital (51, 108, 82)-net over F₅, using

t-expansion [i] based on digital (48, 108, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(51, 51+57, 104)-Net over F₅ — Digital

Digital (51, 108, 104)-net over F₅, using

net from sequence [i] based on digital (51, 103)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 51 and N(F) ≥ 104, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(51, 51+57, 1303)-Net in Base 5 — Upper bound on s

There is no (51, 108, 1304)-net in base 5, because

1 times m-reduction [i] would yield (51, 107, 1304)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 616 533573 102527 503453 521583 630214 207096 606603 782131 397997 354110 152313 465857 > 5¹⁰⁷ [i]