MinT - Best Known (247−93, 247, s)-Nets in Base 4

Best Known (247−93, 247, s)-Nets in Base 4

(247−93, 247, 160)-Net over F₄ — Constructive and digital

Digital (154, 247, 160)-net over F₄, using

3 times m-reduction [i] based on digital (154, 250, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 81, 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, 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]

(247−93, 247, 431)-Net over F₄ — Digital

Digital (154, 247, 431)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−93, 247, 9912)-Net in Base 4 — Upper bound on s

There is no (154, 247, 9913)-net in base 4, because

1 times m-reduction [i] would yield (154, 246, 9913)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12836 081604 369117 454300 387357 117113 787152 623388 118749 929859 273153 935275 501101 931236 055855 928044 053876 532802 937308 621822 975533 072482 189664 400777 892280 > 4²⁴⁶ [i]