MinT - Best Known (69, 69+167, s)-Nets in Base 4

Best Known (69, 69+167, s)-Nets in Base 4

(69, 69+167, 66)-Net over F₄ — Constructive and digital

Digital (69, 236, 66)-net over F₄, using

t-expansion [i] based on digital (49, 236, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(69, 69+167, 99)-Net over F₄ — Digital

Digital (69, 236, 99)-net over F₄, using

t-expansion [i] based on digital (61, 236, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(69, 69+167, 469)-Net in Base 4 — Upper bound on s

There is no (69, 236, 470)-net in base 4, because

1 times m-reduction [i] would yield (69, 235, 470)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3113 189601 806530 005611 473193 897008 293169 197149 113183 564961 101960 899028 982233 469330 942030 064556 596108 088992 737548 440847 984509 241740 546196 236586 > 4²³⁵ [i]