MinT - Best Known (253−95, 253, s)-Nets in Base 4

Best Known (253−95, 253, s)-Nets in Base 4

(253−95, 253, 160)-Net over F₄ — Constructive and digital

Digital (158, 253, 160)-net over F₄, using

t-expansion [i] based on digital (157, 253, 160)-net over F₄, using
- 6 times m-reduction [i] based on digital (157, 259, 160)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (33, 84, 56)-net over F₄, using
      - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ 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]

(253−95, 253, 445)-Net over F₄ — Digital

Digital (158, 253, 445)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−95, 253, 10313)-Net in Base 4 — Upper bound on s

There is no (158, 253, 10314)-net in base 4, because

1 times m-reduction [i] would yield (158, 252, 10314)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 405572 702054 505756 750905 858879 597971 313798 891616 520759 924714 704946 179427 579513 869995 607042 049005 421548 284742 567741 992501 759857 603955 114981 015778 554400 > 4²⁵² [i]