MinT - Best Known (126, 126+98, s)-Nets in Base 2

Best Known (126, 126+98, s)-Nets in Base 2

(126, 126+98, 63)-Net over F₂ — Constructive and digital

Digital (126, 224, 63)-net over F₂, using

4 times m-reduction [i] based on digital (126, 228, 63)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 72, 21)-net over F₂, using
    - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 156, 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]

(126, 126+98, 83)-Net over F₂ — Digital

Digital (126, 224, 83)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+98, 347)-Net over F₂ — Upper bound on s (digital)

There is no digital (126, 224, 348)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²²⁴, 348, F₂, 98) (dual of [348, 124, 99]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(2²²³, 300, S₂, 98), but
    - the linear programming bound shows that MÂ â‰¥ 1 797227 776840 883681 896516 291018 079602 556004 736527 384481 376471 307374 972535 986485 205069 933364 379648 / 112622 624143 760877 566730 935625 > 2²²³ [i]
  2. linear OA(2¹²⁴, 348, F₂, 48) (dual of [348, 224, 49]-code), but
    - discarding factors / shortening the dual code would yield linear OA(2¹²⁴, 332, F₂, 48) (dual of [332, 208, 49]-code), but
      - the improved Johnson bound shows that NÂ â‰¤ 1985 666223 770950 387579 854572 849250 190195 414082 522040 623254 891664 < 2²⁰⁸ [i]

(126, 126+98, 386)-Net in Base 2 — Upper bound on s

There is no (126, 224, 387)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 28 960408 360049 549862 847478 604306 577035 938216 969759 521334 410136 192224 > 2²²⁴ [i]