MinT - Best Known (82, 82+137, s)-Nets in Base 2

Best Known (82, 82+137, s)-Nets in Base 2

(82, 82+137, 51)-Net over F₂ — Constructive and digital

Digital (82, 219, 51)-net over F₂, using

t-expansion [i] based on digital (80, 219, 51)-net over F₂, using
- net from sequence [i] based on digital (80, 50)-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 2 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]

(82, 82+137, 56)-Net over F₂ — Digital

Digital (82, 219, 56)-net over F₂, using

t-expansion [i] based on digital (80, 219, 56)-net over F₂, using
- net from sequence [i] based on digital (80, 55)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 80 and N(F) ≥ 56, using
    - Drinfeld modules of rank 1 [i]

(82, 82+137, 155)-Net in Base 2 — Upper bound on s

There is no (82, 219, 156)-net in base 2, because

1 times m-reduction [i] would yield (82, 218, 156)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 440183 234269 308322 810827 069189 162296 752909 664023 775823 437669 273540 > 2²¹⁸ [i]