MinT - Best Known (139−129, 139, s)-Nets in Base 5

Best Known (139−129, 139, s)-Nets in Base 5

(139−129, 139, 26)-Net over F₅ — Constructive and digital

Digital (10, 139, 26)-net over F₅, using

t-expansion [i] based on digital (9, 139, 26)-net over F₅, using
- net from sequence [i] based on digital (9, 25)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 9 and N(F) ≥ 26, using
    - an explicitly constructive algebraic function field [i]

(139−129, 139, 27)-Net over F₅ — Digital

Digital (10, 139, 27)-net over F₅, using

net from sequence [i] based on digital (10, 26)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 10 and N(F) ≥ 27, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(139−129, 139, 53)-Net in Base 5 — Upper bound on s

There is no (10, 139, 54)-net in base 5, because

34 times m-reduction [i] would yield (10, 105, 54)-net in base 5, but
- extracting embedded OOA [i] would yield OOA(5¹⁰⁵, 54, S₅, 2, 95), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 123 259516 440783 094595 582588 325435 348386 438505 485784 844495 356082 916259 765625 / 4 > 5¹⁰⁵ [i]