MinT - Best Known (242−79, 242, s)-Nets in Base 2

Best Known (242−79, 242, s)-Nets in Base 2

(242−79, 242, 112)-Net over F₂ — Constructive and digital

Digital (163, 242, 112)-net over F₂, using

18 times m-reduction [i] based on digital (163, 260, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 130, 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]

(242−79, 242, 173)-Net over F₂ — Digital

Digital (163, 242, 173)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−79, 242, 1059)-Net in Base 2 — Upper bound on s

There is no (163, 242, 1060)-net in base 2, because

1 times m-reduction [i] would yield (163, 241, 1060)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 624934 855810 546440 751056 812162 190678 558538 370224 668666 366975 397109 345872 > 2²⁴¹ [i]