MinT - Best Known (146−27, 146, s)-Nets in Base 2

Best Known (146−27, 146, s)-Nets in Base 2

(146−27, 146, 265)-Net over F₂ — Constructive and digital

Digital (119, 146, 265)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 14, 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 (105, 132, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 33, 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]

(146−27, 146, 669)-Net over F₂ — Digital

Digital (119, 146, 669)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁴⁶, 669, F₂, 3, 27) (dual of [(669, 3), 1861, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁴⁶, 687, F₂, 3, 27) (dual of [(687, 3), 1915, 28]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(2¹⁴⁶, 2061, F₂, 27) (dual of [2061, 1915, 28]-code), using
    - 1 times code embedding in larger space [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]

(146−27, 146, 12894)-Net in Base 2 — Upper bound on s

There is no (119, 146, 12895)-net in base 2, because

1 times m-reduction [i] would yield (119, 145, 12895)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 44 617408 003796 786264 264174 829223 665719 852964 > 2¹⁴⁵ [i]