MinT - Best Known (137−93, 137, s)-Nets in Base 5

Best Known (137−93, 137, s)-Nets in Base 5

(137−93, 137, 78)-Net over F₅ — Constructive and digital

Digital (44, 137, 78)-net over F₅, using

t-expansion [i] based on digital (38, 137, 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]

(137−93, 137, 84)-Net over F₅ — Digital

Digital (44, 137, 84)-net over F₅, using

t-expansion [i] based on digital (43, 137, 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]

(137−93, 137, 491)-Net in Base 5 — Upper bound on s

There is no (44, 137, 492)-net in base 5, because

1 times m-reduction [i] would yield (44, 136, 492)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 123218 952054 419273 011351 778520 568325 278822 246610 848317 949159 396230 980564 472742 611090 209877 792065 > 5¹³⁶ [i]