MinT - Best Known (131−93, 131, s)-Nets in Base 4

Best Known (131−93, 131, s)-Nets in Base 4

(131−93, 131, 56)-Net over F₄ — Constructive and digital

Digital (38, 131, 56)-net over F₄, using

t-expansion [i] based on digital (33, 131, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(131−93, 131, 66)-Net over F₄ — Digital

Digital (38, 131, 66)-net over F₄, using

t-expansion [i] based on digital (37, 131, 66)-net over F₄, using
- net from sequence [i] based on digital (37, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
    - Drinfeld modules of rank 1 [i]

(131−93, 131, 265)-Net in Base 4 — Upper bound on s

There is no (38, 131, 266)-net in base 4, because

1 times m-reduction [i] would yield (38, 130, 266)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 976436 714554 092163 372770 279623 923833 832710 287408 679602 696941 790105 630601 151104 > 4¹³⁰ [i]