MinT - Best Known (139−77, 139, s)-Nets in Base 5

Best Known (139−77, 139, s)-Nets in Base 5

(139−77, 139, 82)-Net over F₅ — Constructive and digital

Digital (62, 139, 82)-net over F₅, using

t-expansion [i] based on digital (48, 139, 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]

(139−77, 139, 120)-Net over F₅ — Digital

Digital (62, 139, 120)-net over F₅, using

t-expansion [i] based on digital (61, 139, 120)-net over F₅, using
- net from sequence [i] based on digital (61, 119)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 61 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(139−77, 139, 1269)-Net in Base 5 — Upper bound on s

There is no (62, 139, 1270)-net in base 5, because

1 times m-reduction [i] would yield (62, 138, 1270)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 893911 570955 526803 944342 939452 600477 513531 166103 094020 806794 175986 072786 560449 474678 711743 228225 > 5¹³⁸ [i]