MinT - Best Known (10, 10+22, s)-Nets in Base 81

Best Known (10, 10+22, s)-Nets in Base 81

(10, 10+22, 172)-Net over F₈₁ — Constructive and digital

Digital (10, 32, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 32, 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]

(10, 10+22, 244)-Net over F₈₁ — Digital

Digital (10, 32, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 32, 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]

(10, 10+22, 21866)-Net in Base 81 — Upper bound on s

There is no (10, 32, 21867)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 11 793360 759760 445734 748941 698749 856986 248494 021332 831088 618961 > 81³² [i]