Best Known (71, 71+188, s)-Nets in Base 4
(71, 71+188, 66)-Net over F4 — Constructive and digital
Digital (71, 259, 66)-net over F4, using
- t-expansion [i] based on digital (49, 259, 66)-net over F4, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction 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]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
(71, 71+188, 105)-Net over F4 — Digital
Digital (71, 259, 105)-net over F4, using
- t-expansion [i] based on digital (70, 259, 105)-net over F4, using
- net from sequence [i] based on digital (70, 104)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 70 and N(F) ≥ 105, using
- net from sequence [i] based on digital (70, 104)-sequence over F4, using
(71, 71+188, 380)-Net over F4 — Upper bound on s (digital)
There is no digital (71, 259, 381)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4259, 381, F4, 188) (dual of [381, 122, 189]-code), but
- residual code [i] would yield OA(471, 192, S4, 47), but
- 1 times truncation [i] would yield OA(470, 191, S4, 46), but
- the linear programming bound shows that M ≥ 30669 687295 767807 163249 547418 721993 874758 654208 012151 612633 672044 408069 277192 616104 296206 434138 576650 240000 / 20720 048409 770029 043765 568578 931955 503324 140138 849174 037831 490849 > 470 [i]
- 1 times truncation [i] would yield OA(470, 191, S4, 46), but
- residual code [i] would yield OA(471, 192, S4, 47), but
(71, 71+188, 469)-Net in Base 4 — Upper bound on s
There is no (71, 259, 470)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 891897 294710 102498 825888 006652 921579 967529 379058 033011 138631 023443 350684 655003 618258 395353 276864 383252 217757 622931 772219 004523 684609 570228 261674 486592 384064 > 4259 [i]