MinT - Best Known (130, 157, s)-Nets in Base 2

Best Known (130, 157, s)-Nets in Base 2

(130, 157, 320)-Net over F₂ — Constructive and digital

Digital (130, 157, 320)-net over F₂, using

2² times duplication [i] based on digital (128, 155, 320)-net over F₂, using
- t-expansion [i] based on digital (127, 155, 320)-net over F₂, using
  - trace code for nets [i] based on digital (3, 31, 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]

(130, 157, 1024)-Net over F₂ — Digital

Digital (130, 157, 1024)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁵⁷, 1024, F₂, 4, 27) (dual of [(1024, 4), 3939, 28]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2¹⁵⁷, 4096, F₂, 27) (dual of [4096, 3939, 28]-code), using
  - an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 2¹²âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]

(130, 157, 23195)-Net in Base 2 — Upper bound on s

There is no (130, 157, 23196)-net in base 2, because

1 times m-reduction [i] would yield (130, 156, 23196)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 91350 498332 337409 435960 257095 568045 415707 786988 > 2¹⁵⁶ [i]