MinT - Best Known (82, 82+73, s)-Nets in Base 2

Best Known (82, 82+73, s)-Nets in Base 2

(82, 82+73, 51)-Net over F₂ — Constructive and digital

Digital (82, 155, 51)-net over F₂, using

t-expansion [i] based on digital (80, 155, 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+73, 56)-Net over F₂ — Digital

Digital (82, 155, 56)-net over F₂, using

t-expansion [i] based on digital (80, 155, 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+73, 216)-Net over F₂ — Upper bound on s (digital)

There is no digital (82, 155, 217)-net over F₂, because

1 times m-reduction [i] would yield digital (82, 154, 217)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁵⁴, 217, F₂, 72) (dual of [217, 63, 73]-code), but
  - residual code [i] would yield OA(2⁸², 144, S₂, 36), but
    - the linear programming bound shows that MÂ â‰¥ 2893 609949 429043 518839 097243 777165 200630 895720 113073 291264 / 558 874024 262522 794899 259597 123083 > 2⁸² [i]

(82, 82+73, 227)-Net in Base 2 — Upper bound on s

There is no (82, 155, 228)-net in base 2, because

1 times m-reduction [i] would yield (82, 154, 228)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 23531 724905 569682 391212 542040 466419 524063 298256 > 2¹⁵⁴ [i]