MinT - Best Known (179−88, 179, s)-Nets in Base 2

Best Known (179−88, 179, s)-Nets in Base 2

(179−88, 179, 53)-Net over F₂ — Constructive and digital

Digital (91, 179, 53)-net over F₂, using

t-expansion [i] based on digital (90, 179, 53)-net over F₂, using
- net from sequence [i] based on digital (90, 52)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 4 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(179−88, 179, 57)-Net over F₂ — Digital

Digital (91, 179, 57)-net over F₂, using

t-expansion [i] based on digital (83, 179, 57)-net over F₂, using
- net from sequence [i] based on digital (83, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
    - an algebraic function field by Auer [i]

(179−88, 179, 199)-Net over F₂ — Upper bound on s (digital)

There is no digital (91, 179, 200)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹⁷⁹, 200, F₂, 88) (dual of [200, 21, 89]-code), but
- residual code [i] would yield OA(2⁹¹, 111, S₂, 44), but
  - the linear programming bound shows that MÂ â‰¥ 4098 314390 537865 655038 841462 980608 / 1 584999 > 2⁹¹ [i]

(179−88, 179, 229)-Net in Base 2 — Upper bound on s

There is no (91, 179, 230)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 767945 292862 876955 714737 034659 525318 005667 882476 666888 > 2¹⁷⁹ [i]