MinT - Best Known (81, 81+40, s)-Nets in Base 4

Best Known (81, 81+40, s)-Nets in Base 4

(81, 81+40, 195)-Net over F₄ — Constructive and digital

Digital (81, 121, 195)-net over F₄, using

4¹ times duplication [i] based on digital (80, 120, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 40, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(81, 81+40, 196)-Net in Base 4 — Constructive

(81, 121, 196)-net in base 4, using

4¹ times duplication [i] based on (80, 120, 196)-net in base 4, using
- t-expansion [i] based on (79, 120, 196)-net in base 4, using
  - trace code for nets [i] based on (19, 60, 98)-net in base 16, using
    - base change [i] based on digital (7, 48, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(81, 81+40, 374)-Net over F₄ — Digital

Digital (81, 121, 374)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+40, 12135)-Net in Base 4 — Upper bound on s

There is no (81, 121, 12136)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 069177 420399 272962 936628 366591 665283 282879 178335 564007 718308 934077 030126 > 4¹²¹ [i]