MinT - Best Known (229−83, 229, s)-Nets in Base 2

Best Known (229−83, 229, s)-Nets in Base 2

(229−83, 229, 78)-Net over F₂ — Constructive and digital

Digital (146, 229, 78)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (51, 92, 36)-net over F₂, using
  - net from sequence [i] based on digital (51, 35)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
2. digital (54, 137, 42)-net over F₂, using
  - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(229−83, 229, 86)-Net in Base 2 — Constructive

(146, 229, 86)-net in base 2, using

3 times m-reduction [i] based on (146, 232, 86)-net in base 2, using
- trace code for nets [i] based on (30, 116, 43)-net in base 4, using
  - net from sequence [i] based on (30, 42)-sequence in base 4, using
    - base expansion [i] based on digital (60, 42)-sequence over F₂, using
      - t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(229−83, 229, 129)-Net over F₂ — Digital

Digital (146, 229, 129)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(229−83, 229, 703)-Net in Base 2 — Upper bound on s

There is no (146, 229, 704)-net in base 2, because

1 times m-reduction [i] would yield (146, 228, 704)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 452 701972 399594 068665 406334 552462 544222 289181 034156 499192 324194 408975 > 2²²⁸ [i]