MinT - Best Known (123−87, 123, s)-Nets in Base 5

Best Known (123−87, 123, s)-Nets in Base 5

(123−87, 123, 72)-Net over F₅ — Constructive and digital

Digital (36, 123, 72)-net over F₅, using

t-expansion [i] based on digital (31, 123, 72)-net over F₅, using
- net from sequence [i] based on digital (31, 71)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
    - an explicitly constructive algebraic function field [i]

(123−87, 123, 76)-Net over F₅ — Digital

Digital (36, 123, 76)-net over F₅, using

t-expansion [i] based on digital (34, 123, 76)-net over F₅, using
- net from sequence [i] based on digital (34, 75)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 34 and N(F) ≥ 76, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(123−87, 123, 375)-Net in Base 5 — Upper bound on s

There is no (36, 123, 376)-net in base 5, because

1 times m-reduction [i] would yield (36, 122, 376)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 20 672688 458346 047869 300251 215995 561708 413115 600051 403086 838511 615584 620803 330910 118625 > 5¹²² [i]