MinT - Best Known (110−61, 110, s)-Nets in Base 5

Best Known (110−61, 110, s)-Nets in Base 5

(110−61, 110, 82)-Net over F₅ — Constructive and digital

Digital (49, 110, 82)-net over F₅, using

t-expansion [i] based on digital (48, 110, 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]

(110−61, 110, 96)-Net over F₅ — Digital

Digital (49, 110, 96)-net over F₅, using

net from sequence [i] based on digital (49, 95)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 49 and N(F) ≥ 96, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(110−61, 110, 1021)-Net in Base 5 — Upper bound on s

There is no (49, 110, 1022)-net in base 5, because

1 times m-reduction [i] would yield (49, 109, 1022)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 15743 554441 255922 331597 310852 348067 722893 250125 832199 048451 837902 893588 011585 > 5¹⁰⁹ [i]