MinT - Best Known (57, 57+77, s)-Nets in Base 5

Best Known (57, 57+77, s)-Nets in Base 5

(57, 57+77, 82)-Net over F₅ — Constructive and digital

Digital (57, 134, 82)-net over F₅, using

t-expansion [i] based on digital (48, 134, 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]

(57, 57+77, 112)-Net over F₅ — Digital

Digital (57, 134, 112)-net over F₅, using

net from sequence [i] based on digital (57, 111)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 57 and N(F) ≥ 112, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(57, 57+77, 1022)-Net in Base 5 — Upper bound on s

There is no (57, 134, 1023)-net in base 5, because

1 times m-reduction [i] would yield (57, 133, 1023)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 947 404024 273693 556612 946392 328284 592964 850660 723157 068793 519538 326793 234855 474717 902641 790825 > 5¹³³ [i]