MinT - Best Known (71, 71+180, s)-Nets in Base 4

Best Known (71, 71+180, s)-Nets in Base 4

(71, 71+180, 66)-Net over F₄ — Constructive and digital

Digital (71, 251, 66)-net over F₄, using

t-expansion [i] based on digital (49, 251, 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]

(71, 71+180, 105)-Net over F₄ — Digital

Digital (71, 251, 105)-net over F₄, using

t-expansion [i] based on digital (70, 251, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(71, 71+180, 409)-Net over F₄ — Upper bound on s (digital)

There is no digital (71, 251, 410)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁵¹, 410, F₄, 180) (dual of [410, 159, 181]-code), but
- residual code [i] would yield OA(4⁷¹, 229, S₄, 45), but
  - the linear programming bound shows that MÂ â‰¥ 877 912702 924905 625680 913013 034326 550228 995916 587809 875578 704494 502776 185113 709858 393246 285168 640000 / 155 079140 374101 434134 168939 696571 367080 998053 409783 569671 > 4⁷¹ [i]

(71, 71+180, 475)-Net in Base 4 — Upper bound on s

There is no (71, 251, 476)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 15 124046 920660 820107 210126 323218 538305 087999 366581 439584 061826 478449 627589 388676 194619 705829 654766 349460 246293 325080 418044 767639 378846 124801 386127 028609 > 4²⁵¹ [i]