MinT - Best Known (231−42, 231, s)-Nets in Base 2

Best Known (231−42, 231, s)-Nets in Base 2

(231−42, 231, 320)-Net over F₂ — Constructive and digital

Digital (189, 231, 320)-net over F₂, using

2¹ times duplication [i] based on digital (188, 230, 320)-net over F₂, using
- t-expansion [i] based on digital (187, 230, 320)-net over F₂, using
  - trace code for nets [i] based on digital (3, 46, 64)-net over F₃₂, using
    - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
        a function field by SÃ©mirat [i]

(231−42, 231, 846)-Net over F₂ — Digital

Digital (189, 231, 846)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³¹, 846, F₂, 2, 42) (dual of [(846, 2), 1461, 43]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²³¹, 1023, F₂, 2, 42) (dual of [(1023, 2), 1815, 43]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2²³¹, 2046, F₂, 42) (dual of [2046, 1815, 43]-code), using
    - discarding factors / shortening the dual code based on linear OA(2²³¹, 2047, F₂, 42) (dual of [2047, 1816, 43]-code), using
      - 1 times truncation [i] based on linear OA(2²³², 2048, F₂, 43) (dual of [2048, 1816, 44]-code), using
        an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]

(231−42, 231, 17744)-Net in Base 2 — Upper bound on s

There is no (189, 231, 17745)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3452 695854 840284 739350 210889 116578 517495 670326 513377 237769 096898 833472 > 2²³¹ [i]