MinT - Best Known (58, 58+73, s)-Nets in Base 5

Best Known (58, 58+73, s)-Nets in Base 5

(58, 58+73, 82)-Net over F₅ — Constructive and digital

Digital (58, 131, 82)-net over F₅, using

t-expansion [i] based on digital (48, 131, 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]

(58, 58+73, 112)-Net over F₅ — Digital

Digital (58, 131, 112)-net over F₅, using

t-expansion [i] based on digital (57, 131, 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]

(58, 58+73, 1166)-Net in Base 5 — Upper bound on s

There is no (58, 131, 1167)-net in base 5, because

1 times m-reduction [i] would yield (58, 130, 1167)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 7 353737 127559 026569 943819 060390 832661 575352 663527 703441 464866 172103 515079 505579 892700 842353 > 5¹³⁰ [i]