MinT - Best Known (88, 88+23, s)-Nets in Base 2

Best Known (88, 88+23, s)-Nets in Base 2

(88, 88+23, 195)-Net over F₂ — Constructive and digital

Digital (88, 111, 195)-net over F₂, using

trace code for nets [i] based on digital (14, 37, 65)-net over F₈, using
- net from sequence [i] based on digital (14, 64)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
    - the Suzuki function field over F₈ [i]

(88, 88+23, 341)-Net over F₂ — Digital

Digital (88, 111, 341)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹¹¹, 341, F₂, 3, 23) (dual of [(341, 3), 912, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2¹¹¹, 1023, F₂, 23) (dual of [1023, 912, 24]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [0,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(88, 88+23, 5011)-Net in Base 2 — Upper bound on s

There is no (88, 111, 5012)-net in base 2, because

1 times m-reduction [i] would yield (88, 110, 5012)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1300 680253 099749 673158 113517 372594 > 2¹¹⁰ [i]