MinT - Best Known (171−91, 171, s)-Nets in Base 2

Best Known (171−91, 171, s)-Nets in Base 2

(171−91, 171, 51)-Net over F₂ — Constructive and digital

Digital (80, 171, 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]

(171−91, 171, 56)-Net over F₂ — Digital

Digital (80, 171, 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]

(171−91, 171, 170)-Net over F₂ — Upper bound on s (digital)

There is no digital (80, 171, 171)-net over F₂, because

11 times m-reduction [i] would yield digital (80, 160, 171)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁶⁰, 171, F₂, 80) (dual of [171, 11, 81]-code), but
  - residual code [i] would yield linear OA(2⁸⁰, 90, F₂, 40) (dual of [90, 10, 41]-code), but
    - residual code [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]

(171−91, 171, 171)-Net in Base 2 — Upper bound on s

There is no (80, 171, 172)-net in base 2, because

5 times m-reduction [i] would yield (80, 166, 172)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁶⁶, 172, S₂, 86), but
  - adding a parity check bit [i] would yield OA(2¹⁶⁷, 173, S₂, 87), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 2244 866514 940266 882360 859903 052210 718191 512386 011136 / 11 > 2¹⁶⁷ [i]