MinT - Best Known (206−113, 206, s)-Nets in Base 2

Best Known (206−113, 206, s)-Nets in Base 2

(206−113, 206, 53)-Net over F₂ — Constructive and digital

Digital (93, 206, 53)-net over F₂, using

t-expansion [i] based on digital (90, 206, 53)-net over F₂, using
- net from sequence [i] based on digital (90, 52)-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 4 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]

(206−113, 206, 60)-Net over F₂ — Digital

Digital (93, 206, 60)-net over F₂, using

t-expansion [i] based on digital (92, 206, 60)-net over F₂, using
- net from sequence [i] based on digital (92, 59)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 92 and N(F) ≥ 60, using
    - Drinfeld modules of rank 1 [i]

(206−113, 206, 195)-Net over F₂ — Upper bound on s (digital)

There is no digital (93, 206, 196)-net over F₂, because

17 times m-reduction [i] would yield digital (93, 189, 196)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁸⁹, 196, F₂, 96) (dual of [196, 7, 97]-code), but
  - Griesmer bound [i]

(206−113, 206, 198)-Net in Base 2 — Upper bound on s

There is no (93, 206, 199)-net in base 2, because

13 times m-reduction [i] would yield (93, 193, 199)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁹³, 199, S₂, 100), but
  - adding a parity check bit [i] would yield OA(2¹⁹⁴, 200, S₂, 101), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 1 606938 044258 990275 541962 092341 162602 522202 993782 792835 301376 / 51 > 2¹⁹⁴ [i]