MinT - Best Known (58, 58+87, s)-Nets in Base 5

Best Known (58, 58+87, s)-Nets in Base 5

(58, 58+87, 82)-Net over F₅ — Constructive and digital

Digital (58, 145, 82)-net over F₅, using

t-expansion [i] based on digital (48, 145, 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+87, 112)-Net over F₅ — Digital

Digital (58, 145, 112)-net over F₅, using

t-expansion [i] based on digital (57, 145, 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+87, 893)-Net in Base 5 — Upper bound on s

There is no (58, 145, 894)-net in base 5, because

1 times m-reduction [i] would yield (58, 144, 894)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 45599 844956 574813 952796 190177 063314 953277 964471 234805 146205 189833 007836 594652 236344 171605 766715 843865 > 5¹⁴⁴ [i]