MinT - Best Known (137, 137+94, s)-Nets in Base 4

Best Known (137, 137+94, s)-Nets in Base 4

(137, 137+94, 137)-Net over F₄ — Constructive and digital

Digital (137, 231, 137)-net over F₄, using

4 times m-reduction [i] based on digital (137, 235, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 64, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 171, 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]

(137, 137+94, 314)-Net over F₄ — Digital

Digital (137, 231, 314)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+94, 5533)-Net in Base 4 — Upper bound on s

There is no (137, 231, 5534)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 929456 756274 128019 416138 844523 794877 092628 757716 814245 320464 065106 300023 773484 542258 484210 267426 470638 695584 591083 524135 642213 423369 638560 > 4²³¹ [i]