MinT - Best Known (49, 49+89, s)-Nets in Base 5

Best Known (49, 49+89, s)-Nets in Base 5

(49, 49+89, 82)-Net over F₅ — Constructive and digital

Digital (49, 138, 82)-net over F₅, using

t-expansion [i] based on digital (48, 138, 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]

(49, 49+89, 96)-Net over F₅ — Digital

Digital (49, 138, 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]

(49, 49+89, 615)-Net in Base 5 — Upper bound on s

There is no (49, 138, 616)-net in base 5, because

1 times m-reduction [i] would yield (49, 137, 616)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 593441 080736 884480 137244 361802 721575 357855 527032 718019 608137 180229 515001 051327 942296 788171 367425 > 5¹³⁷ [i]