MinT - Best Known (123−67, 123, s)-Nets in Base 5

Best Known (123−67, 123, s)-Nets in Base 5

(123−67, 123, 82)-Net over F₅ — Constructive and digital

Digital (56, 123, 82)-net over F₅, using

t-expansion [i] based on digital (48, 123, 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]

(123−67, 123, 108)-Net over F₅ — Digital

Digital (56, 123, 108)-net over F₅, using

t-expansion [i] based on digital (55, 123, 108)-net over F₅, using
- net from sequence [i] based on digital (55, 107)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 55 and N(F) ≥ 108, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(123−67, 123, 1238)-Net in Base 5 — Upper bound on s

There is no (56, 123, 1239)-net in base 5, because

1 times m-reduction [i] would yield (56, 122, 1239)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 18 904695 045585 558253 807177 041660 644680 537275 953895 628938 836370 955058 049425 305721 647837 > 5¹²² [i]