MinT - Best Known (168−31, 168, s)-Nets in Base 2

Best Known (168−31, 168, s)-Nets in Base 2

(168−31, 168, 269)-Net over F₂ — Constructive and digital

Digital (137, 168, 269)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 20, 9)-net over F₂, using
  - net from sequence [i] based on digital (5, 8)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 5 and N(F) ≥ 9, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (117, 148, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 37, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(168−31, 168, 687)-Net over F₂ — Digital

Digital (137, 168, 687)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁶⁸, 687, F₂, 3, 31) (dual of [(687, 3), 1893, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2¹⁶⁸, 2061, F₂, 31) (dual of [2061, 1893, 32]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(2¹⁶⁷, 2060, F₂, 31) (dual of [2060, 1893, 32]-code), using
    - constructionÂ X applied to C_e(30) âŠ‚ C_e(28) [i] based on
      1. linear OA(2¹⁶⁶, 2048, F₂, 31) (dual of [2048, 1882, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      2. linear OA(2¹⁵⁵, 2048, F₂, 29) (dual of [2048, 1893, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      3. linear OA(2¹, 12, F₂, 1) (dual of [12, 11, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹, s, F₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(168−31, 168, 14406)-Net in Base 2 — Upper bound on s

There is no (137, 168, 14407)-net in base 2, because

1 times m-reduction [i] would yield (137, 167, 14407)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 187 084166 359053 144309 118680 220352 044835 158861 257840 > 2¹⁶⁷ [i]