MinT - Best Known (119−21, 119, s)-Nets in Base 2

Best Known (119−21, 119, s)-Nets in Base 2

(119−21, 119, 265)-Net over F₂ — Constructive and digital

Digital (98, 119, 265)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 5)-net over F₂, using
  - net from sequence [i] based on digital (1, 4)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (87, 108, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 27, 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]

(119−21, 119, 693)-Net over F₂ — Digital

Digital (98, 119, 693)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹¹⁹, 693, F₂, 3, 21) (dual of [(693, 3), 1960, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2¹¹⁹, 2079, F₂, 21) (dual of [2079, 1960, 22]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(2¹¹⁷, 2077, F₂, 21) (dual of [2077, 1960, 22]-code), using
    - constructionÂ X applied to C([0,10]) âŠ‚ C([0,8]) [i] based on
      1. linear OA(2¹¹¹, 2049, F₂, 21) (dual of [2049, 1938, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049Â | 2²²âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
      2. linear OA(2⁸⁹, 2049, F₂, 17) (dual of [2049, 1960, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049Â | 2²²âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
      3. linear OA(2⁶, 28, F₂, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,2)), using
        discarding factors / shortening the dual code based on linear OA(2⁶, 32, F₂, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
        Reedâ€“Muller code RM(1,5) [i]
        caps in base bÂ =Â 2 [i]

(119−21, 119, 16133)-Net in Base 2 — Upper bound on s

There is no (98, 119, 16134)-net in base 2, because

1 times m-reduction [i] would yield (98, 118, 16134)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 332320 611820 311149 493485 600164 465728 > 2¹¹⁸ [i]