MinT - Best Known (105−57, 105, s)-Nets in Base 5

Best Known (105−57, 105, s)-Nets in Base 5

(105−57, 105, 82)-Net over F₅ — Constructive and digital

Digital (48, 105, 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]

(105−57, 105, 92)-Net over F₅ — Digital

Digital (48, 105, 92)-net over F₅, using

t-expansion [i] based on digital (47, 105, 92)-net over F₅, using
- net from sequence [i] based on digital (47, 91)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 47 and N(F) ≥ 92, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(105−57, 105, 1094)-Net in Base 5 — Upper bound on s

There is no (48, 105, 1095)-net in base 5, because

1 times m-reduction [i] would yield (48, 104, 1095)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 5 026049 927819 303302 520724 440432 731552 418612 058490 967410 544565 443212 063377 > 5¹⁰⁴ [i]