MinT - Best Known (68−58, 68, s)-Nets in Base 81

Best Known (68−58, 68, s)-Nets in Base 81

(68−58, 68, 172)-Net over F₈₁ — Constructive and digital

Digital (10, 68, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 68, 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]

(68−58, 68, 244)-Net over F₈₁ — Digital

Digital (10, 68, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 68, 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]

(68−58, 68, 4341)-Net in Base 81 — Upper bound on s

There is no (10, 68, 4342)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 5992 947721 990837 452953 884662 939152 699715 975681 991421 351255 618772 192137 952525 672691 611134 046391 721604 611532 134006 811659 768734 540641 > 81⁶⁸ [i]