Best Known (208−152, 208, s)-Nets in Base 4
(208−152, 208, 66)-Net over F4 — Constructive and digital
Digital (56, 208, 66)-net over F4, using
- t-expansion [i] based on digital (49, 208, 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
(208−152, 208, 91)-Net over F4 — Digital
Digital (56, 208, 91)-net over F4, using
- t-expansion [i] based on digital (50, 208, 91)-net over F4, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 50 and N(F) ≥ 91, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
(208−152, 208, 292)-Net over F4 — Upper bound on s (digital)
There is no digital (56, 208, 293)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4208, 293, F4, 152) (dual of [293, 85, 153]-code), but
- residual code [i] would yield OA(456, 140, S4, 38), but
- the linear programming bound shows that M ≥ 33224 510089 193434 121103 982150 072093 135568 373830 671910 070349 424562 458616 197168 496640 000000 / 5 929310 526700 508468 903034 465148 531898 302680 222550 630653 > 456 [i]
- residual code [i] would yield OA(456, 140, S4, 38), but
(208−152, 208, 371)-Net in Base 4 — Upper bound on s
There is no (56, 208, 372)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 178217 613911 442436 563516 070212 126993 474044 969678 242839 644897 480478 821948 043145 667584 761850 704369 277074 540292 656558 199433 002612 > 4208 [i]