MinT - Best Known (53, 109, s)-Nets in Base 2

Best Known (53, 109, s)-Nets in Base 2

(53, 109, 36)-Net over F₂ — Constructive and digital

Digital (53, 109, 36)-net over F₂, using

t-expansion [i] based on digital (51, 109, 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]

(53, 109, 40)-Net over F₂ — Digital

Digital (53, 109, 40)-net over F₂, using

t-expansion [i] based on digital (50, 109, 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]

(53, 109, 116)-Net over F₂ — Upper bound on s (digital)

There is no digital (53, 109, 117)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹⁰⁹, 117, F₂, 56) (dual of [117, 8, 57]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(2¹¹⁰, 118, F₂, 56) (dual of [118, 8, 57]-code), but
  - adding a parity check bit [i] would yield linear OA(2¹¹¹, 119, F₂, 57) (dual of [119, 8, 58]-code), but
    - â€œDMaâ€ bound on codes from Brouwerâ€™s database [i]

(53, 109, 117)-Net in Base 2 — Upper bound on s

There is no (53, 109, 118)-net in base 2, because

2 times m-reduction [i] would yield (53, 107, 118)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁰⁷, 118, S₂, 54), but
  - the linear programming bound shows that MÂ â‰¥ 6490 371073 168534 535663 120411 525120 / 33 > 2¹⁰⁷ [i]