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

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

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

Digital (51, 102, 82)-net over F₅, using

t-expansion [i] based on digital (48, 102, 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]

(51, 51+51, 106)-Net over F₅ — Digital

Digital (51, 102, 106)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+51, 1677)-Net in Base 5 — Upper bound on s

There is no (51, 102, 1678)-net in base 5, because

1 times m-reduction [i] would yield (51, 101, 1678)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 39595 979583 905943 373520 797720 055389 167685 139351 205487 950988 017344 690233 > 5¹⁰¹ [i]