MinT - Best Known (63−31, 63, s)-Nets in Base 3

Best Known (63−31, 63, s)-Nets in Base 3

(63−31, 63, 38)-Net over F₃ — Constructive and digital

Digital (32, 63, 38)-net over F₃, using

net from sequence [i] based on digital (32, 37)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
  - an explicitly constructive algebraic function field [i]

(63−31, 63, 42)-Net over F₃ — Digital

Digital (32, 63, 42)-net over F₃, using

t-expansion [i] based on digital (29, 63, 42)-net over F₃, using
- net from sequence [i] based on digital (29, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 29 and N(F) ≥ 42, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(63−31, 63, 286)-Net in Base 3 — Upper bound on s

There is no (32, 63, 287)-net in base 3, because

1 times m-reduction [i] would yield (32, 62, 287)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 382027 460887 939725 968130 593891 > 3⁶² [i]