MinT - Best Known (155, 256, s)-Nets in Base 2

Best Known (155, 256, s)-Nets in Base 2

(155, 256, 78)-Net over F₂ — Constructive and digital

Digital (155, 256, 78)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (51, 101, 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, 155, 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]

(155, 256, 84)-Net in Base 2 — Constructive

(155, 256, 84)-net in base 2, using

trace code for nets [i] based on (27, 128, 42)-net in base 4, using
- net from sequence [i] based on (27, 41)-sequence in base 4, using
  - base expansion [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]

(155, 256, 118)-Net over F₂ — Digital

Digital (155, 256, 118)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(155, 256, 597)-Net in Base 2 — Upper bound on s

There is no (155, 256, 598)-net in base 2, because

1 times m-reduction [i] would yield (155, 255, 598)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 61033 023845 290266 318236 224485 680062 857262 253055 369861 957226 677930 560761 165524 > 2²⁵⁵ [i]