MinT - Best Known (82, 158, s)-Nets in Base 2

Best Known (82, 158, s)-Nets in Base 2

(82, 158, 51)-Net over F₂ — Constructive and digital

Digital (82, 158, 51)-net over F₂, using

t-expansion [i] based on digital (80, 158, 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]

(82, 158, 56)-Net over F₂ — Digital

Digital (82, 158, 56)-net over F₂, using

t-expansion [i] based on digital (80, 158, 56)-net over F₂, using
- net from sequence [i] based on digital (80, 55)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 80 and N(F) ≥ 56, using
    - Drinfeld modules of rank 1 [i]

(82, 158, 189)-Net over F₂ — Upper bound on s (digital)

There is no digital (82, 158, 190)-net over F₂, because

2 times m-reduction [i] would yield digital (82, 156, 190)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁵⁶, 190, F₂, 74) (dual of [190, 34, 75]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(2¹⁵⁵, 178, F₂, 74) (dual of [178, 23, 75]-code), but
      - adding a parity check bit [i] would yield linear OA(2¹⁵⁶, 179, F₂, 75) (dual of [179, 23, 76]-code), but
        â€œBKâ€ bound on codes from Brouwerâ€™s database [i]
    2. OA(2³⁴, 190, S₂, 12), but
      - discarding factors would yield OA(2³⁴, 154, S₂, 12), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 17486 314616 > 2³⁴ [i]

(82, 158, 216)-Net in Base 2 — Upper bound on s

There is no (82, 158, 217)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 388102 958415 665324 150660 051198 683953 226955 148200 > 2¹⁵⁸ [i]