MinT - Best Known (229−149, 229, s)-Nets in Base 4

Best Known (229−149, 229, s)-Nets in Base 4

(229−149, 229, 104)-Net over F₄ — Constructive and digital

Digital (80, 229, 104)-net over F₄, using

t-expansion [i] based on digital (73, 229, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(229−149, 229, 112)-Net over F₄ — Digital

Digital (80, 229, 112)-net over F₄, using

t-expansion [i] based on digital (73, 229, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(229−149, 229, 618)-Net in Base 4 — Upper bound on s

There is no (80, 229, 619)-net in base 4, because

1 times m-reduction [i] would yield (80, 228, 619)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 202220 262166 221042 481919 997292 202872 070102 573107 016153 635842 008671 463725 023361 016437 747914 879923 348260 278669 926852 380728 049057 582225 587091 > 4²²⁸ [i]