MinT - Best Known (126−81, 126, s)-Nets in Base 5

Best Known (126−81, 126, s)-Nets in Base 5

(126−81, 126, 78)-Net over F₅ — Constructive and digital

Digital (45, 126, 78)-net over F₅, using

t-expansion [i] based on digital (38, 126, 78)-net over F₅, using
- net from sequence [i] based on digital (38, 77)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 38 and N(F) ≥ 78, using
    - an explicitly constructive algebraic function field [i]

(126−81, 126, 88)-Net over F₅ — Digital

Digital (45, 126, 88)-net over F₅, using

net from sequence [i] based on digital (45, 87)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 45 and N(F) ≥ 88, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(126−81, 126, 573)-Net in Base 5 — Upper bound on s

There is no (45, 126, 574)-net in base 5, because

1 times m-reduction [i] would yield (45, 125, 574)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2417 542348 582336 904934 175153 375107 505154 408410 866190 710006 953594 059233 118400 819832 237953 > 5¹²⁵ [i]