MinT - Best Known (81, 81+18, s)-Nets in Base 2

Best Known (81, 81+18, s)-Nets in Base 2

(81, 81+18, 260)-Net over F₂ — Constructive and digital

Digital (81, 99, 260)-net over F₂, using

1 times m-reduction [i] based on digital (81, 100, 260)-net over F₂, using
- trace code for nets [i] based on digital (6, 25, 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]

(81, 81+18, 681)-Net over F₂ — Digital

Digital (81, 99, 681)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁹⁹, 681, F₂, 3, 18) (dual of [(681, 3), 1944, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2⁹⁹, 682, F₂, 3, 18) (dual of [(682, 3), 1947, 19]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(2⁹⁹, 2046, F₂, 18) (dual of [2046, 1947, 19]-code), using
    - discarding factors / shortening the dual code based on linear OA(2⁹⁹, 2047, F₂, 18) (dual of [2047, 1948, 19]-code), using
      - 1 times truncation [i] based on 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]

(81, 81+18, 8480)-Net in Base 2 — Upper bound on s

There is no (81, 99, 8481)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 634224 769080 046593 551464 007768 > 2⁹⁹ [i]