MinT - Best Known (144−26, 144, s)-Nets in Base 2

Best Known (144−26, 144, s)-Nets in Base 2

(144−26, 144, 267)-Net over F₂ — Constructive and digital

Digital (118, 144, 267)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 16, 7)-net over F₂, using
  - net from sequence [i] based on digital (3, 6)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 3 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (102, 128, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 32, 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]

(144−26, 144, 686)-Net over F₂ — Digital

Digital (118, 144, 686)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁴⁴, 686, F₂, 3, 26) (dual of [(686, 3), 1914, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2¹⁴⁴, 2058, F₂, 26) (dual of [2058, 1914, 27]-code), using
  - discarding factors / shortening the dual code based on linear OA(2¹⁴⁴, 2059, F₂, 26) (dual of [2059, 1915, 27]-code), using
    - 1 times truncation [i] based on linear OA(2¹⁴⁵, 2060, F₂, 27) (dual of [2060, 1915, 28]-code), using
      - constructionÂ X applied to C_e(26) âŠ‚ C_e(24) [i] based on
        linear OA(2¹⁴⁴, 2048, F₂, 27) (dual of [2048, 1904, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
        linear OA(2¹³³, 2048, F₂, 25) (dual of [2048, 1915, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
        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]

(144−26, 144, 12224)-Net in Base 2 — Upper bound on s

There is no (118, 144, 12225)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 22 321158 035159 520285 387429 622587 969048 111776 > 2¹⁴⁴ [i]