MinT - Best Known (253−105, 253, s)-Nets in Base 4

Best Known (253−105, 253, s)-Nets in Base 4

(253−105, 253, 138)-Net over F₄ — Constructive and digital

Digital (148, 253, 138)-net over F₄, using

3 times m-reduction [i] based on digital (148, 256, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 75, 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, 181, 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]

(253−105, 253, 320)-Net over F₄ — Digital

Digital (148, 253, 320)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−105, 253, 5534)-Net in Base 4 — Upper bound on s

There is no (148, 253, 5535)-net in base 4, because

1 times m-reduction [i] would yield (148, 252, 5535)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 403121 332447 669539 884005 295866 556715 166259 443584 754974 743532 301945 139034 832072 428406 784820 639470 032619 354664 075164 944491 994595 410181 288768 542587 551940 > 4²⁵² [i]