MinT - Best Known (209−90, 209, s)-Nets in Base 2

Best Known (209−90, 209, s)-Nets in Base 2

(209−90, 209, 62)-Net over F₂ — Constructive and digital

Digital (119, 209, 62)-net over F₂, using

2 times m-reduction [i] based on digital (119, 211, 62)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (19, 65, 20)-net over F₂, using
    - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 146, 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]

(209−90, 209, 81)-Net over F₂ — Digital

Digital (119, 209, 81)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−90, 209, 358)-Net over F₂ — Upper bound on s (digital)

There is no digital (119, 209, 359)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁰⁹, 359, F₂, 90) (dual of [359, 150, 91]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(2²⁰⁸, 299, S₂, 90), but
    - adding a parity check bit [i] would yield OA(2²⁰⁹, 300, S₂, 91), but
      - the linear programming bound shows that MÂ â‰¥ 437987 836379 661851 428112 482969 460104 960459 291237 979833 080590 515802 667419 590325 215760 903409 709975 863296 / 433 753365 263741 276278 643266 060465 678125 > 2²⁰⁹ [i]
  2. linear OA(2¹⁵⁰, 359, F₂, 60) (dual of [359, 209, 61]-code), but
    - discarding factors / shortening the dual code would yield linear OA(2¹⁵⁰, 354, F₂, 60) (dual of [354, 204, 61]-code), but
      - the improved Johnson bound shows that NÂ â‰¤ 570 693273 077464 953259 984159 736256 114541 311702 095039 925212 853150 < 2²⁰⁴ [i]

(209−90, 209, 378)-Net in Base 2 — Upper bound on s

There is no (119, 209, 379)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 897 750651 213333 649356 162003 931970 929177 531355 213656 732703 603152 > 2²⁰⁹ [i]