MinT - Best Known (69, 69+109, s)-Nets in Base 4

Best Known (69, 69+109, s)-Nets in Base 4

(69, 69+109, 66)-Net over F₄ — Constructive and digital

Digital (69, 178, 66)-net over F₄, using

t-expansion [i] based on digital (49, 178, 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+109, 99)-Net over F₄ — Digital

Digital (69, 178, 99)-net over F₄, using

t-expansion [i] based on digital (61, 178, 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+109, 614)-Net in Base 4 — Upper bound on s

There is no (69, 178, 615)-net in base 4, because

1 times m-reduction [i] would yield (69, 177, 615)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 39746 676903 158589 973804 227432 363945 266742 369734 953274 469920 255358 367575 504386 958543 116652 344508 023597 021356 > 4¹⁷⁷ [i]