MinT - Best Known (56, 56+68, s)-Nets in Base 5

Best Known (56, 56+68, s)-Nets in Base 5

(56, 56+68, 82)-Net over F₅ — Constructive and digital

Digital (56, 124, 82)-net over F₅, using

t-expansion [i] based on digital (48, 124, 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+68, 108)-Net over F₅ — Digital

Digital (56, 124, 108)-net over F₅, using

t-expansion [i] based on digital (55, 124, 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+68, 1173)-Net in Base 5 — Upper bound on s

There is no (56, 124, 1174)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 475 833913 911721 495902 037057 877635 214572 213971 004171 689984 116555 020634 158025 129540 614785 > 5¹²⁴ [i]