MinT - Best Known (139−93, 139, s)-Nets in Base 5

Best Known (139−93, 139, s)-Nets in Base 5

(139−93, 139, 78)-Net over F₅ — Constructive and digital

Digital (46, 139, 78)-net over F₅, using

t-expansion [i] based on digital (38, 139, 78)-net over F₅, using
- net from sequence [i] based on digital (38, 77)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 38 and N(F) ≥ 78, using
    - an explicitly constructive algebraic function field [i]

(139−93, 139, 88)-Net over F₅ — Digital

Digital (46, 139, 88)-net over F₅, using

t-expansion [i] based on digital (45, 139, 88)-net over F₅, using
- net from sequence [i] based on digital (45, 87)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 45 and N(F) ≥ 88, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(139−93, 139, 529)-Net in Base 5 — Upper bound on s

There is no (46, 139, 530)-net in base 5, because

1 times m-reduction [i] would yield (46, 138, 530)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3 075282 843175 896315 437421 460512 770983 478965 689023 096843 301991 950477 754981 863134 134283 046871 601025 > 5¹³⁸ [i]