MinT - Best Known (109, 109+24, s)-Nets in Base 2

Best Known (109, 109+24, s)-Nets in Base 2

(109, 109+24, 265)-Net over F₂ — Constructive and digital

Digital (109, 133, 265)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 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 (96, 120, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 30, 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]

(109, 109+24, 686)-Net over F₂ — Digital

Digital (109, 133, 686)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹³³, 686, F₂, 3, 24) (dual of [(686, 3), 1925, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2¹³³, 2058, F₂, 24) (dual of [2058, 1925, 25]-code), using
  - discarding factors / shortening the dual code based on linear OA(2¹³³, 2059, F₂, 24) (dual of [2059, 1926, 25]-code), using
    - 1 times truncation [i] based on linear OA(2¹³⁴, 2060, F₂, 25) (dual of [2060, 1926, 26]-code), using
      - constructionÂ X applied to C_e(24) âŠ‚ C_e(22) [i] based on
        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(2¹²², 2048, F₂, 23) (dual of [2048, 1926, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [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]

(109, 109+24, 11458)-Net in Base 2 — Upper bound on s

There is no (109, 133, 11459)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 10898 610050 701868 281971 866840 701764 364752 > 2¹³³ [i]