MinT - Best Known (36−21, 36, s)-Nets in Base 2

Best Known (36−21, 36, s)-Nets in Base 2

(36−21, 36, 17)-Net over F₂ — Constructive and digital

Digital (15, 36, 17)-net over F₂, using

net from sequence [i] based on digital (15, 16)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
  - an explicitly constructive algebraic function field [i]

(36−21, 36, 37)-Net over F₂ — Upper bound on s (digital)

There is no digital (15, 36, 38)-net over F₂, because

5 times m-reduction [i] would yield digital (15, 31, 38)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2³¹, 38, F₂, 16) (dual of [38, 7, 17]-code), but
  - residual code [i] would yield linear OA(2¹⁵, 21, F₂, 8) (dual of [21, 6, 9]-code), but
    - residual code [i] would yield linear OA(2⁷, 12, F₂, 4) (dual of [12, 5, 5]-code), but
      - bound by Fontaine and Peterson [i]

(36−21, 36, 38)-Net in Base 2 — Upper bound on s

There is no (15, 36, 39)-net in base 2, because

1 times m-reduction [i] would yield (15, 35, 39)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 39104 534028 > 2³⁵ [i]
- extracting embedded orthogonal array [i] would yield OA(2³⁵, 39, S₂, 20), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 274877 906944 / 7 > 2³⁵ [i]