MinT - Best Known (213−96, 213, s)-Nets in Base 2

Best Known (213−96, 213, s)-Nets in Base 2

(213−96, 213, 59)-Net over F₂ — Constructive and digital

Digital (117, 213, 59)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (15, 63, 17)-net over F₂, using
  - net from sequence [i] based on digital (15, 16)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (54, 150, 42)-net over F₂, using
  - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(213−96, 213, 74)-Net over F₂ — Digital

Digital (117, 213, 74)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−96, 213, 292)-Net in Base 2 — Upper bound on s

There is no (117, 213, 293)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2²¹³, 293, S₂, 96), but
- 6 times code embedding in larger space [i] would yield OA(2²¹⁹, 299, S₂, 96), but
  - adding a parity check bit [i] would yield OA(2²²⁰, 300, S₂, 97), but
    - the linear programming bound shows that MÂ â‰¥ 2 707506 179671 220537 436761 186277 555172 982275 381427 765331 317661 637213 928123 767152 572244 071765 637582 553088 / 1 434637 915544 533146 412054 075529 090625 > 2²²⁰ [i]