MinT - Best Known (52, 52+63, s)-Nets in Base 5

Best Known (52, 52+63, s)-Nets in Base 5

(52, 52+63, 82)-Net over F₅ — Constructive and digital

Digital (52, 115, 82)-net over F₅, using

t-expansion [i] based on digital (48, 115, 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+63, 104)-Net over F₅ — Digital

Digital (52, 115, 104)-net over F₅, using

t-expansion [i] based on digital (51, 115, 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+63, 1131)-Net in Base 5 — Upper bound on s

There is no (52, 115, 1132)-net in base 5, because

1 times m-reduction [i] would yield (52, 114, 1132)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 48 318314 102259 061081 076699 261786 751089 229337 589165 768911 576360 731063 674617 363505 > 5¹¹⁴ [i]