Best Known (249−181, 249, s)-Nets in Base 4
(249−181, 249, 66)-Net over F4 — Constructive and digital
Digital (68, 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
- 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
(249−181, 249, 99)-Net over F4 — Digital
Digital (68, 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
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 61 and N(F) ≥ 99, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
(249−181, 249, 360)-Net over F4 — Upper bound on s (digital)
There is no digital (68, 249, 361)-net over F4, because
- 1 times m-reduction [i] would yield digital (68, 248, 361)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4248, 361, F4, 180) (dual of [361, 113, 181]-code), but
- residual code [i] would yield OA(468, 180, S4, 45), but
- the linear programming bound shows that M ≥ 328494 790735 001198 305199 602445 532392 471737 863161 499929 581656 031112 774235 621273 901469 693549 435944 960000 000000 / 3 747617 146367 023709 191641 153229 380578 240235 288823 915002 005290 401803 > 468 [i]
- residual code [i] would yield OA(468, 180, S4, 45), but
- extracting embedded orthogonal array [i] would yield linear OA(4248, 361, F4, 180) (dual of [361, 113, 181]-code), but
(249−181, 249, 450)-Net in Base 4 — Upper bound on s
There is no (68, 249, 451)-net in base 4, because
- 1 times m-reduction [i] would yield (68, 248, 451)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 218055 118198 373581 516951 194040 423242 282304 957321 797875 795348 111343 912405 837881 013305 928711 494390 791994 334260 944252 838350 431380 324153 301666 947921 061328 > 4248 [i]