MinT - Best Known (66−37, 66, s)-Nets in Base 4

Best Known (66−37, 66, s)-Nets in Base 4

(66−37, 66, 36)-Net over F₄ — Constructive and digital

Digital (29, 66, 36)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 22, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (7, 44, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]

(66−37, 66, 42)-Net in Base 4 — Constructive

(29, 66, 42)-net in base 4, using

t-expansion [i] based on (27, 66, 42)-net in base 4, using
- net from sequence [i] based on (27, 41)-sequence in base 4, using
  - base expansion [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(66−37, 66, 55)-Net over F₄ — Digital

Digital (29, 66, 55)-net over F₄, using

t-expansion [i] based on digital (26, 66, 55)-net over F₄, using
- net from sequence [i] based on digital (26, 54)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 26 and N(F) ≥ 55, using
    - Drinfeld modules of rank 1 [i]

(66−37, 66, 361)-Net in Base 4 — Upper bound on s

There is no (29, 66, 362)-net in base 4, because

1 times m-reduction [i] would yield (29, 65, 362)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1384 206497 175797 199058 706078 936425 066360 > 4⁶⁵ [i]