MinT - Best Known (85−44, 85, s)-Nets in Base 2

Best Known (85−44, 85, s)-Nets in Base 2

(85−44, 85, 33)-Net over F₂ — Constructive and digital

Digital (41, 85, 33)-net over F₂, using

t-expansion [i] based on digital (39, 85, 33)-net over F₂, using
- net from sequence [i] based on digital (39, 32)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(85−44, 85, 92)-Net over F₂ — Upper bound on s (digital)

There is no digital (41, 85, 93)-net over F₂, because

2 times m-reduction [i] would yield digital (41, 83, 93)-net over F₂, but
- extracting embedded orthogonal array [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]

(85−44, 85, 93)-Net in Base 2 — Upper bound on s

There is no (41, 85, 94)-net in base 2, because

2 times m-reduction [i] would yield (41, 83, 94)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁸³, 94, S₂, 42), but
  - the linear programming bound shows that MÂ â‰¥ 15628 992995 977925 970601 443328 / 1287 > 2⁸³ [i]