MinT - Best Known (131−87, 131, s)-Nets in Base 5

Best Known (131−87, 131, s)-Nets in Base 5

(131−87, 131, 78)-Net over F₅ — Constructive and digital

Digital (44, 131, 78)-net over F₅, using

t-expansion [i] based on digital (38, 131, 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]

(131−87, 131, 84)-Net over F₅ — Digital

Digital (44, 131, 84)-net over F₅, using

t-expansion [i] based on digital (43, 131, 84)-net over F₅, using
- net from sequence [i] based on digital (43, 83)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 43 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(131−87, 131, 516)-Net in Base 5 — Upper bound on s

There is no (44, 131, 517)-net in base 5, because

1 times m-reduction [i] would yield (44, 130, 517)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 7 565646 282875 281919 111183 586872 398717 239944 559630 945068 620120 567507 820522 898575 784434 206125 > 5¹³⁰ [i]