MinT - Best Known (56, 56+87, s)-Nets in Base 5

Best Known (56, 56+87, s)-Nets in Base 5

(56, 56+87, 82)-Net over F₅ — Constructive and digital

Digital (56, 143, 82)-net over F₅, using

t-expansion [i] based on digital (48, 143, 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]

(56, 56+87, 108)-Net over F₅ — Digital

Digital (56, 143, 108)-net over F₅, using

t-expansion [i] based on digital (55, 143, 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]

(56, 56+87, 826)-Net in Base 5 — Upper bound on s

There is no (56, 143, 827)-net in base 5, because

1 times m-reduction [i] would yield (56, 142, 827)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1797 902000 859078 180995 696994 292665 882343 404531 641027 795270 624408 512498 625597 089624 570943 252735 950725 > 5¹⁴² [i]