MinT - Best Known (194, 194+43, s)-Nets in Base 2

Best Known (194, 194+43, s)-Nets in Base 2

(194, 194+43, 320)-Net over F₂ — Constructive and digital

Digital (194, 237, 320)-net over F₂, using

2² times duplication [i] based on digital (192, 235, 320)-net over F₂, using
- t-expansion [i] based on digital (191, 235, 320)-net over F₂, using
  - trace code for nets [i] based on digital (3, 47, 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]

(194, 194+43, 869)-Net over F₂ — Digital

Digital (194, 237, 869)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³⁷, 869, F₂, 2, 43) (dual of [(869, 2), 1501, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²³⁷, 1032, F₂, 2, 43) (dual of [(1032, 2), 1827, 44]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(2²³³, 1030, F₂, 2, 43) (dual of [(1030, 2), 1827, 44]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(2²³³, 2060, F₂, 43) (dual of [2060, 1827, 44]-code), using
      - constructionÂ X applied to C_e(42) âŠ‚ C_e(40) [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]
        linear OA(2²²¹, 2048, F₂, 41) (dual of [2048, 1827, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [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]

(194, 194+43, 20934)-Net in Base 2 — Upper bound on s

There is no (194, 237, 20935)-net in base 2, because

1 times m-reduction [i] would yield (194, 236, 20935)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 110528 483785 356097 919643 677265 463646 254702 943113 666888 028996 847520 675888 > 2²³⁶ [i]