MinT - Best Known (140−77, 140, s)-Nets in Base 5

Best Known (140−77, 140, s)-Nets in Base 5

(140−77, 140, 82)-Net over F₅ — Constructive and digital

Digital (63, 140, 82)-net over F₅, using

t-expansion [i] based on digital (48, 140, 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]

(140−77, 140, 120)-Net over F₅ — Digital

Digital (63, 140, 120)-net over F₅, using

t-expansion [i] based on digital (61, 140, 120)-net over F₅, using
- net from sequence [i] based on digital (61, 119)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 61 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(140−77, 140, 1325)-Net in Base 5 — Upper bound on s

There is no (63, 140, 1326)-net in base 5, because

1 times m-reduction [i] would yield (63, 139, 1326)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 14 413679 407487 537219 526139 289190 328119 107127 522254 431852 881635 337556 174513 474385 852673 184100 671425 > 5¹³⁹ [i]