MinT - Best Known (228−117, 228, s)-Nets in Base 4

Best Known (228−117, 228, s)-Nets in Base 4

(228−117, 228, 130)-Net over F₄ — Constructive and digital

Digital (111, 228, 130)-net over F₄, using

t-expansion [i] based on digital (105, 228, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(228−117, 228, 165)-Net over F₄ — Digital

Digital (111, 228, 165)-net over F₄, using

t-expansion [i] based on digital (109, 228, 165)-net over F₄, using
- net from sequence [i] based on digital (109, 164)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
    - Drinfeld modules of rank 1 [i]

(228−117, 228, 1652)-Net in Base 4 — Upper bound on s

There is no (111, 228, 1653)-net in base 4, because

1 times m-reduction [i] would yield (111, 227, 1653)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47069 180136 760983 622546 008704 126814 612228 846361 760644 128122 453385 602678 816199 186201 068262 019245 766509 263545 843278 664539 562944 951494 506000 > 4²²⁷ [i]