MinT - Best Known (254−143, 254, s)-Nets in Base 2

Best Known (254−143, 254, s)-Nets in Base 2

(254−143, 254, 57)-Net over F₂ — Constructive and digital

Digital (111, 254, 57)-net over F₂, using

t-expansion [i] based on digital (110, 254, 57)-net over F₂, using
- net from sequence [i] based on digital (110, 56)-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 8 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]

(254−143, 254, 72)-Net over F₂ — Digital

Digital (111, 254, 72)-net over F₂, using

t-expansion [i] based on digital (110, 254, 72)-net over F₂, using
- net from sequence [i] based on digital (110, 71)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 110 and N(F) ≥ 72, using
    - Drinfeld modules of rank 1 [i]

(254−143, 254, 229)-Net in Base 2 — Upper bound on s

There is no (111, 254, 230)-net in base 2, because

1 times m-reduction [i] would yield (111, 253, 230)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 14751 477139 696094 715490 449154 195003 674455 242046 100558 682497 240136 645587 499836 > 2²⁵³ [i]