MinT - Best Known (81, 81+45, s)-Nets in Base 4

Best Known (81, 81+45, s)-Nets in Base 4

(81, 81+45, 140)-Net over F₄ — Constructive and digital

Digital (81, 126, 140)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 24, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (57, 102, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 51, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(81, 81+45, 152)-Net in Base 4 — Constructive

(81, 126, 152)-net in base 4, using

trace code for nets [i] based on (18, 63, 76)-net in base 16, using
- 2 times m-reduction [i] based on (18, 65, 76)-net in base 16, using
  - base change [i] based on digital (5, 52, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(81, 81+45, 290)-Net over F₄ — Digital

Digital (81, 126, 290)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+45, 7935)-Net in Base 4 — Upper bound on s

There is no (81, 126, 7936)-net in base 4, because

1 times m-reduction [i] would yield (81, 125, 7936)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1813 698899 681325 289500 496371 654460 405642 733545 233126 478321 111518 467678 790001 > 4¹²⁵ [i]