MinT - Best Known (238−96, 238, s)-Nets in Base 4

Best Known (238−96, 238, s)-Nets in Base 4

(238−96, 238, 138)-Net over F₄ — Constructive and digital

Digital (142, 238, 138)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 69, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 169, 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]

(238−96, 238, 332)-Net over F₄ — Digital

Digital (142, 238, 332)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(238−96, 238, 5998)-Net in Base 4 — Upper bound on s

There is no (142, 238, 5999)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 195959 604857 341070 819294 260531 114600 113825 470334 136487 175396 392328 356913 584125 069388 865463 555631 023485 474539 045200 648823 398469 601127 713254 794372 > 4²³⁸ [i]