MinT - Best Known (135, 135+111, s)-Nets in Base 2

Best Known (135, 135+111, s)-Nets in Base 2

(135, 135+111, 66)-Net over F₂ — Constructive and digital

Digital (135, 246, 66)-net over F₂, using

3 times m-reduction [i] based on digital (135, 249, 66)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 96, 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]
  2. digital (39, 153, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂ (see above)

(135, 135+111, 83)-Net over F₂ — Digital

Digital (135, 246, 83)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+111, 331)-Net over F₂ — Upper bound on s (digital)

There is no digital (135, 246, 332)-net over F₂, because

1 times m-reduction [i] would yield digital (135, 245, 332)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁴⁵, 332, F₂, 110) (dual of [332, 87, 111]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(2²⁴⁴, 300, S₂, 110), but
      - the linear programming bound shows that MÂ â‰¥ 727209 932038 995964 963694 786156 506719 769259 430085 807852 102717 045283 079365 167115 254635 995296 956416 / 21378 491075 472167 578125 > 2²⁴⁴ [i]
    2. OA(2⁸⁷, 332, S₂, 32), but
      - discarding factors would yield OA(2⁸⁷, 302, S₂, 32), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 161 699122 225452 699910 750634 > 2⁸⁷ [i]

(135, 135+111, 391)-Net in Base 2 — Upper bound on s

There is no (135, 246, 392)-net in base 2, because

1 times m-reduction [i] would yield (135, 245, 392)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 57 493903 038656 840184 024525 583036 262005 180426 565596 031606 162786 708987 748360 > 2²⁴⁵ [i]