MinT - Best Known (81, 81+21, s)-Nets in Base 2

Best Known (81, 81+21, s)-Nets in Base 2

(81, 81+21, 180)-Net over F₂ — Constructive and digital

Digital (81, 102, 180)-net over F₂, using

2² times duplication [i] based on digital (79, 100, 180)-net over F₂, using
- trace code for nets [i] based on digital (4, 25, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]

(81, 81+21, 345)-Net over F₂ — Digital

Digital (81, 102, 345)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁰², 345, F₂, 3, 21) (dual of [(345, 3), 933, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2¹⁰², 1035, F₂, 21) (dual of [1035, 933, 22]-code), using
  - constructionÂ X applied to C_e(20) âŠ‚ C_e(18) [i] based on
    1. linear OA(2¹⁰¹, 1024, F₂, 21) (dual of [1024, 923, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    2. linear OA(2⁹¹, 1024, F₂, 19) (dual of [1024, 933, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    3. linear OA(2¹, 11, F₂, 1) (dual of [11, 10, 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]

(81, 81+21, 4955)-Net in Base 2 — Upper bound on s

There is no (81, 102, 4956)-net in base 2, because

1 times m-reduction [i] would yield (81, 101, 4956)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2 536420 344083 891286 609647 703093 > 2¹⁰¹ [i]