MinT - Best Known (83, 168, s)-Nets in Base 2

Best Known (83, 168, s)-Nets in Base 2

(83, 168, 51)-Net over F₂ — Constructive and digital

Digital (83, 168, 51)-net over F₂, using

t-expansion [i] based on digital (80, 168, 51)-net over F₂, using
- net from sequence [i] based on digital (80, 50)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 2 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(83, 168, 57)-Net over F₂ — Digital

Digital (83, 168, 57)-net over F₂, using

net from sequence [i] based on digital (83, 56)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
  - an algebraic function field by Auer [i]

(83, 168, 177)-Net over F₂ — Upper bound on s (digital)

There is no digital (83, 168, 178)-net over F₂, because

1 times m-reduction [i] would yield digital (83, 167, 178)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁶⁷, 178, F₂, 84) (dual of [178, 11, 85]-code), but
  - residual code [i] would yield linear OA(2⁸³, 93, F₂, 42) (dual of [93, 10, 43]-code), but
    - residual code [i] would yield linear OA(2⁴¹, 50, F₂, 21) (dual of [50, 9, 22]-code), but
      - 1 times truncation [i] would yield linear OA(2⁴⁰, 49, F₂, 20) (dual of [49, 9, 21]-code), but
        â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(83, 168, 203)-Net in Base 2 — Upper bound on s

There is no (83, 168, 204)-net in base 2, because

1 times m-reduction [i] would yield (83, 167, 204)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 204 727916 359907 742101 899783 607952 602857 929818 334750 > 2¹⁶⁷ [i]