MinT - Best Known (77−66, 77, s)-Nets in Base 81

Best Known (77−66, 77, s)-Nets in Base 81

(77−66, 77, 172)-Net over F₈₁ — Constructive and digital

Digital (11, 77, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 77, 172)-net over F₈₁, using
- net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
    - a function field by SÃ©mirat [i]

(77−66, 77, 244)-Net over F₈₁ — Digital

Digital (11, 77, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 77, 244)-net over F₈₁, using
- net from sequence [i] based on digital (9, 243)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 9 and N(F) ≥ 244, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₈₁ [i]

(77−66, 77, 4654)-Net in Base 81 — Upper bound on s

There is no (11, 77, 4655)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 901 656136 772480 856479 460580 702881 256101 646659 841699 771281 177958 888252 161127 307607 200185 633180 588843 934602 358258 063480 324516 540466 848410 689191 937201 > 81⁷⁷ [i]