MinT - Best Known (123−59, 123, s)-Nets in Base 5

Best Known (123−59, 123, s)-Nets in Base 5

(123−59, 123, 90)-Net over F₅ — Constructive and digital

Digital (64, 123, 90)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 33, 18)-net over F₅, using
  - net from sequence [i] based on digital (4, 17)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 90, 72)-net over F₅, using
  - net from sequence [i] based on digital (31, 71)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
      - an explicitly constructive algebraic function field [i]

(123−59, 123, 142)-Net over F₅ — Digital

Digital (64, 123, 142)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−59, 123, 2523)-Net in Base 5 — Upper bound on s

There is no (64, 123, 2524)-net in base 5, because

1 times m-reduction [i] would yield (64, 122, 2524)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 19 013866 889117 204664 880409 631349 581164 414466 740794 368465 326331 427343 202819 137978 704625 > 5¹²² [i]