MinT - Best Known (51, 51+63, s)-Nets in Base 5

Best Known (51, 51+63, s)-Nets in Base 5

(51, 51+63, 82)-Net over F₅ — Constructive and digital

Digital (51, 114, 82)-net over F₅, using

t-expansion [i] based on digital (48, 114, 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+63, 104)-Net over F₅ — Digital

Digital (51, 114, 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+63, 1073)-Net in Base 5 — Upper bound on s

There is no (51, 114, 1074)-net in base 5, because

1 times m-reduction [i] would yield (51, 113, 1074)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 9 775104 430855 504075 678891 385212 751797 842377 636784 208073 452409 132516 459241 364201 > 5¹¹³ [i]