MinT - Best Known (137, 238, s)-Nets in Base 4

Best Known (137, 238, s)-Nets in Base 4

(137, 238, 134)-Net over F₄ — Constructive and digital

Digital (137, 238, 134)-net over F₄, using

1 times m-reduction [i] based on digital (137, 239, 134)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 64, 30)-net over F₄, using
    - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
        F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 175, 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, 238, 282)-Net over F₄ — Digital

Digital (137, 238, 282)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 238, 4596)-Net in Base 4 — Upper bound on s

There is no (137, 238, 4597)-net in base 4, because

1 times m-reduction [i] would yield (137, 237, 4597)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48938 857246 808012 574996 361848 891600 239608 392133 627957 674093 529904 470108 445123 661524 768056 429128 711412 883510 311753 061100 753278 898999 276415 672930 > 4²³⁷ [i]