MinT - Best Known (81−70, 81, s)-Nets in Base 81

Best Known (81−70, 81, s)-Nets in Base 81

(81−70, 81, 172)-Net over F₈₁ — Constructive and digital

Digital (11, 81, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 81, 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]

(81−70, 81, 244)-Net over F₈₁ — Digital

Digital (11, 81, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 81, 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]

(81−70, 81, 4521)-Net in Base 81 — Upper bound on s

There is no (11, 81, 4522)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 38810 136526 613506 502535 016473 185643 105884 211639 533857 691712 573751 994617 537054 061984 091949 511001 674023 989549 871888 926905 376872 019608 057112 927607 266929 045601 > 81⁸¹ [i]