Best Known (65, 249, s)-Nets in Base 4
(65, 249, 66)-Net over F4 — Constructive and digital
Digital (65, 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
(65, 249, 99)-Net over F4 — Digital
Digital (65, 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
(65, 249, 297)-Net over F4 — Upper bound on s (digital)
There is no digital (65, 249, 298)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4249, 298, F4, 184) (dual of [298, 49, 185]-code), but
- residual code [i] would yield OA(465, 113, S4, 46), but
- the linear programming bound shows that M ≥ 145 402369 805359 268193 209685 952643 209942 587568 332064 303419 732807 154505 932977 868145 129044 024267 389501 535040 862760 730624 / 100423 807776 117676 644806 780906 988983 236254 180840 898357 650546 636091 023031 034405 > 465 [i]
- residual code [i] would yield OA(465, 113, S4, 46), but
(65, 249, 425)-Net in Base 4 — Upper bound on s
There is no (65, 249, 426)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 890331 195481 600730 047243 947013 050710 926870 163970 140090 168293 435194 756459 882959 959253 222639 127925 842712 662595 720244 452534 885424 613330 339597 474006 042640 > 4249 [i]