MinT - Best Known (143−71, 143, s)-Nets in Base 4

Best Known (143−71, 143, s)-Nets in Base 4

(143−71, 143, 71)-Net over F₄ — Constructive and digital

Digital (72, 143, 71)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 39, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (33, 104, 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]

(143−71, 143, 110)-Net over F₄ — Digital

Digital (72, 143, 110)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(143−71, 143, 1256)-Net in Base 4 — Upper bound on s

There is no (72, 143, 1257)-net in base 4, because

1 times m-reduction [i] would yield (72, 142, 1257)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 31 742203 005220 621826 367950 878290 675171 111745 055211 181066 404687 124229 941147 573324 210120 > 4¹⁴² [i]