Best Known (255−185, 255, s)-Nets in Base 4
(255−185, 255, 66)-Net over F4 — Constructive and digital
Digital (70, 255, 66)-net over F4, using
- t-expansion [i] based on digital (49, 255, 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
(255−185, 255, 105)-Net over F4 — Digital
Digital (70, 255, 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
(255−185, 255, 375)-Net over F4 — Upper bound on s (digital)
There is no digital (70, 255, 376)-net over F4, because
- 1 times m-reduction [i] would yield digital (70, 254, 376)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4254, 376, F4, 184) (dual of [376, 122, 185]-code), but
- residual code [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]
- residual code [i] would yield OA(470, 191, S4, 46), but
- extracting embedded orthogonal array [i] would yield linear OA(4254, 376, F4, 184) (dual of [376, 122, 185]-code), but
(255−185, 255, 464)-Net in Base 4 — Upper bound on s
There is no (70, 255, 465)-net in base 4, because
- 1 times m-reduction [i] would yield (70, 254, 465)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 960 976936 397345 410183 317013 616966 741198 167742 393957 016871 019354 379881 135959 422965 808628 581814 626370 637302 118770 531852 809116 924805 718072 141663 254292 434400 > 4254 [i]