MinT - Best Known (172−79, 172, s)-Nets in Base 3

Best Known (172−79, 172, s)-Nets in Base 3

(172−79, 172, 74)-Net over F₃ — Constructive and digital

Digital (93, 172, 74)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 66, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]
2. digital (27, 106, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃ (see above)

(172−79, 172, 111)-Net over F₃ — Digital

Digital (93, 172, 111)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−79, 172, 913)-Net in Base 3 — Upper bound on s

There is no (93, 172, 914)-net in base 3, because

1 times m-reduction [i] would yield (93, 171, 914)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3935 580999 575556 737441 676812 837557 443970 169013 092246 258125 083253 311108 945656 653545 > 3¹⁷¹ [i]