MinT - Best Known (111, 111+25, s)-Nets in Base 2

Best Known (111, 111+25, s)-Nets in Base 2

(111, 111+25, 263)-Net over F₂ — Constructive and digital

Digital (111, 136, 263)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 12, 3)-net over F₂, using
  - net from sequence [i] based on digital (0, 2)-sequence over F₂, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 0 and N(F) ≥ 3, using
      - the rational function field F₂(x) [i]
    - Niederreiter sequence [i]
    - Sobol sequence [i]
2. digital (99, 124, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 31, 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]

(111, 111+25, 670)-Net over F₂ — Digital

Digital (111, 136, 670)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹³⁶, 670, F₂, 3, 25) (dual of [(670, 3), 1874, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹³⁶, 687, F₂, 3, 25) (dual of [(687, 3), 1925, 26]-NRT-code), using
  - 2¹ times duplication [i] based on linear OOA(2¹³⁵, 687, F₂, 3, 25) (dual of [(687, 3), 1926, 26]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(2¹³⁵, 2061, F₂, 25) (dual of [2061, 1926, 26]-code), using
      - 1 times code embedding in larger space [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]

(111, 111+25, 12863)-Net in Base 2 — Upper bound on s

There is no (111, 136, 12864)-net in base 2, because

1 times m-reduction [i] would yield (111, 135, 12864)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 43576 862727 045512 969705 680825 057656 523737 > 2¹³⁵ [i]

Best Known (111, 111+25, s)-Nets in Base 2

(111, 111+25, 263)-Net over F2 — Constructive and digital

(111, 111+25, 670)-Net over F2 — Digital

(111, 111+25, 12863)-Net in Base 2 — Upper bound on s

(111, 111+25, 263)-Net over F₂ — Constructive and digital

(111, 111+25, 670)-Net over F₂ — Digital