MinT - Best Known (127−29, 127, s)-Nets in Base 2

Best Known (127−29, 127, s)-Nets in Base 2

(127−29, 127, 144)-Net over F₂ — Constructive and digital

Digital (98, 127, 144)-net over F₂, using

t-expansion [i] based on digital (97, 127, 144)-net over F₂, using
- 2 times m-reduction [i] based on digital (97, 129, 144)-net over F₂, using
  - trace code for nets [i] based on digital (11, 43, 48)-net over F₈, using
    - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
        a function field by SÃ©mirat [i]

(127−29, 127, 256)-Net over F₂ — Digital

Digital (98, 127, 256)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹²⁷, 256, F₂, 2, 29) (dual of [(256, 2), 385, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2¹²⁷, 512, F₂, 29) (dual of [512, 385, 30]-code), using
  - an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 511Â = 2⁹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(127−29, 127, 3075)-Net in Base 2 — Upper bound on s

There is no (98, 127, 3076)-net in base 2, because

1 times m-reduction [i] would yield (98, 126, 3076)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 85 441075 813466 170345 666358 204722 119424 > 2¹²⁶ [i]