MinT - Best Known (56, 56+51, s)-Nets in Base 5

Best Known (56, 56+51, s)-Nets in Base 5

(56, 56+51, 82)-Net over F₅ — Constructive and digital

Digital (56, 107, 82)-net over F₅, using

t-expansion [i] based on digital (48, 107, 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+51, 129)-Net over F₅ — Digital

Digital (56, 107, 129)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+51, 2321)-Net in Base 5 — Upper bound on s

There is no (56, 107, 2322)-net in base 5, because

1 times m-reduction [i] would yield (56, 106, 2322)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 123 644258 480013 832658 446878 024849 992558 062488 196930 697343 590502 306755 516233 > 5¹⁰⁶ [i]