MinT - Best Known (69, 69+180, s)-Nets in Base 4

Best Known (69, 69+180, s)-Nets in Base 4

(69, 69+180, 66)-Net over F₄ — Constructive and digital

Digital (69, 249, 66)-net over F₄, using

t-expansion [i] based on digital (49, 249, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(69, 69+180, 99)-Net over F₄ — Digital

Digital (69, 249, 99)-net over F₄, using

t-expansion [i] based on digital (61, 249, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(69, 69+180, 375)-Net over F₄ — Upper bound on s (digital)

There is no digital (69, 249, 376)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁹, 376, F₄, 180) (dual of [376, 127, 181]-code), but
- residual code [i] would yield OA(4⁶⁹, 195, S₄, 45), but
  - the linear programming bound shows that MÂ â‰¥ 82 828010 425088 398135 365726 677314 464149 340113 774380 345347 664035 522400 039552 947596 743692 557981 180235 874304 000000 / 227 578582 237535 291818 752893 814619 068871 044126 923591 235904 712727 792773 > 4⁶⁹ [i]

(69, 69+180, 458)-Net in Base 4 — Upper bound on s

There is no (69, 249, 459)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 865613 934990 415674 642318 278576 717069 579509 633708 442532 909296 922724 312106 900526 411546 906319 282185 073487 169338 144531 217513 397899 507637 415089 480313 327656 > 4²⁴⁹ [i]