MinT - Best Known (105, 105+23, s)-Nets in Base 2

Best Known (105, 105+23, s)-Nets in Base 2

(105, 105+23, 265)-Net over F₂ — Constructive and digital

Digital (105, 128, 265)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 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 (93, 116, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 29, 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]

(105, 105+23, 692)-Net over F₂ — Digital

Digital (105, 128, 692)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹²⁸, 692, F₂, 3, 23) (dual of [(692, 3), 1948, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2¹²⁸, 2076, F₂, 23) (dual of [2076, 1948, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(18) [i] based on
    1. 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]
    2. linear OA(2¹⁰⁰, 2048, F₂, 19) (dual of [2048, 1948, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [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]

(105, 105+23, 14657)-Net in Base 2 — Upper bound on s

There is no (105, 128, 14658)-net in base 2, because

1 times m-reduction [i] would yield (105, 127, 14658)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 170 167020 281909 318572 337279 565088 134592 > 2¹²⁷ [i]