MinT - Best Known (57, 57+53, s)-Nets in Base 5

Best Known (57, 57+53, s)-Nets in Base 5

(57, 57+53, 82)-Net over F₅ — Constructive and digital

Digital (57, 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]

(57, 57+53, 127)-Net over F₅ — Digital

Digital (57, 110, 127)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+53, 2227)-Net in Base 5 — Upper bound on s

There is no (57, 110, 2228)-net in base 5, because

1 times m-reduction [i] would yield (57, 109, 2228)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 15464 354218 360157 181307 174462 704513 926656 982486 173520 324702 021432 144591 852993 > 5¹⁰⁹ [i]