MinT - Best Known (52, 52+57, s)-Nets in Base 5

Best Known (52, 52+57, s)-Nets in Base 5

(52, 52+57, 82)-Net over F₅ — Constructive and digital

Digital (52, 109, 82)-net over F₅, using

t-expansion [i] based on digital (48, 109, 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]

(52, 52+57, 104)-Net over F₅ — Digital

Digital (52, 109, 104)-net over F₅, using

t-expansion [i] based on digital (51, 109, 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]

(52, 52+57, 1382)-Net in Base 5 — Upper bound on s

There is no (52, 109, 1383)-net in base 5, because

1 times m-reduction [i] would yield (52, 108, 1383)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3122 932128 856792 295196 854140 789018 106886 559839 744933 465071 251961 475434 606225 > 5¹⁰⁸ [i]