MinT - Best Known (142−71, 142, s)-Nets in Base 4

Best Known (142−71, 142, s)-Nets in Base 4

(142−71, 142, 70)-Net over F₄ — Constructive and digital

Digital (71, 142, 70)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 38, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, 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]

(142−71, 142, 107)-Net over F₄ — Digital

Digital (71, 142, 107)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(142−71, 142, 1206)-Net in Base 4 — Upper bound on s

There is no (71, 142, 1207)-net in base 4, because

1 times m-reduction [i] would yield (71, 141, 1207)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 915071 320869 939406 285972 642965 803507 359671 300764 641597 896406 663454 098386 652534 775040 > 4¹⁴¹ [i]