MinT - Best Known (229−111, 229, s)-Nets in Base 2

Best Known (229−111, 229, s)-Nets in Base 2

(229−111, 229, 57)-Net over F₂ — Constructive and digital

Digital (118, 229, 57)-net over F₂, using

t-expansion [i] based on digital (110, 229, 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]

(229−111, 229, 73)-Net over F₂ — Digital

Digital (118, 229, 73)-net over F₂, using

t-expansion [i] based on digital (114, 229, 73)-net over F₂, using
- net from sequence [i] based on digital (114, 72)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 114 and N(F) ≥ 73, using
    - Drinfeld modules of rank 1 [i]

(229−111, 229, 250)-Net in Base 2 — Upper bound on s

There is no (118, 229, 251)-net in base 2, because

3 times m-reduction [i] would yield (118, 226, 251)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²⁶, 251, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 3 068587 541377 441989 862758 184092 770418 630025 105253 803106 088322 664550 769029 545984 / 21915 413575 > 2²²⁶ [i]