MinT - Best Known (71, 71+21, s)-Nets in Base 2

Best Known (71, 71+21, s)-Nets in Base 2

(71, 71+21, 135)-Net over F₂ — Constructive and digital

Digital (71, 92, 135)-net over F₂, using

1 times m-reduction [i] based on digital (71, 93, 135)-net over F₂, using
- trace code for nets [i] based on digital (9, 31, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]

(71, 71+21, 217)-Net over F₂ — Digital

Digital (71, 92, 217)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁹², 217, F₂, 2, 21) (dual of [(217, 2), 342, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2⁹², 261, F₂, 2, 21) (dual of [(261, 2), 430, 22]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2⁹², 522, F₂, 21) (dual of [522, 430, 22]-code), using
    - constructionÂ X applied to C_e(20) âŠ‚ C_e(18) [i] based on
      1. linear OA(2⁹¹, 512, F₂, 21) (dual of [512, 421, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 511Â = 2⁹âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      2. linear OA(2⁸², 512, F₂, 19) (dual of [512, 430, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 511Â = 2⁹âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
      3. linear OA(2¹, 10, F₂, 1) (dual of [10, 9, 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]

(71, 71+21, 2470)-Net in Base 2 — Upper bound on s

There is no (71, 92, 2471)-net in base 2, because

1 times m-reduction [i] would yield (71, 91, 2471)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2479 183500 489497 956187 163036 > 2⁹¹ [i]