MinT - Best Known (52, 52+60, s)-Nets in Base 2

Best Known (52, 52+60, s)-Nets in Base 2

(52, 52+60, 36)-Net over F₂ — Constructive and digital

Digital (52, 112, 36)-net over F₂, using

t-expansion [i] based on digital (51, 112, 36)-net over F₂, using
- net from sequence [i] based on digital (51, 35)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(52, 52+60, 40)-Net over F₂ — Digital

Digital (52, 112, 40)-net over F₂, using

t-expansion [i] based on digital (50, 112, 40)-net over F₂, using
- net from sequence [i] based on digital (50, 39)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 50 and N(F) ≥ 40, using
    - Drinfeld modules of rank 1 [i]

(52, 52+60, 113)-Net over F₂ — Upper bound on s (digital)

There is no digital (52, 112, 114)-net over F₂, because

4 times m-reduction [i] would yield digital (52, 108, 114)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁰⁸, 114, F₂, 56) (dual of [114, 6, 57]-code), but
  - Griesmer bound [i]

(52, 52+60, 114)-Net in Base 2 — Upper bound on s

There is no (52, 112, 115)-net in base 2, because

2 times m-reduction [i] would yield (52, 110, 115)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹¹⁰, 115, S₂, 58), but
  - adding a parity check bit [i] would yield OA(2¹¹¹, 116, S₂, 59), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 41538 374868 278621 028243 970633 760768 / 15 > 2¹¹¹ [i]