MinT - Best Known (77−67, 77, s)-Nets in Base 81

Best Known (77−67, 77, s)-Nets in Base 81

(77−67, 77, 172)-Net over F₈₁ — Constructive and digital

Digital (10, 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]

(77−67, 77, 244)-Net over F₈₁ — Digital

Digital (10, 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]

(77−67, 77, 4072)-Net in Base 81 — Upper bound on s

There is no (10, 77, 4073)-net in base 81, because

1 times m-reduction [i] would yield (10, 76, 4073)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 11 169910 316599 362579 548207 217472 395235 334375 553055 329660 302562 437017 416483 345624 266494 347751 083014 975614 165269 259499 059632 023391 923188 919627 571921 > 81⁷⁶ [i]