MinT - Best Known (238−97, 238, s)-Nets in Base 4

Best Known (238−97, 238, s)-Nets in Base 4

(238−97, 238, 137)-Net over F₄ — Constructive and digital

Digital (141, 238, 137)-net over F₄, using

9 times m-reduction [i] based on digital (141, 247, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 68, 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, 179, 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]

(238−97, 238, 321)-Net over F₄ — Digital

Digital (141, 238, 321)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(238−97, 238, 5826)-Net in Base 4 — Upper bound on s

There is no (141, 238, 5827)-net in base 4, because

1 times m-reduction [i] would yield (141, 237, 5827)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48949 307947 538195 666594 565930 673476 907414 917966 055991 326568 497119 757325 062867 332380 530078 216208 295604 685651 262114 484928 002973 835951 269389 081924 > 4²³⁷ [i]