Best Known (55, 55+156, s)-Nets in Base 4
(55, 55+156, 66)-Net over F4 — Constructive and digital
Digital (55, 211, 66)-net over F4, using
- t-expansion [i] based on digital (49, 211, 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
(55, 55+156, 91)-Net over F4 — Digital
Digital (55, 211, 91)-net over F4, using
- t-expansion [i] based on digital (50, 211, 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
(55, 55+156, 260)-Net over F4 — Upper bound on s (digital)
There is no digital (55, 211, 261)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4211, 261, F4, 156) (dual of [261, 50, 157]-code), but
- residual code [i] would yield OA(455, 104, S4, 39), but
- the linear programming bound shows that M ≥ 7 723473 632829 389285 976355 541453 254660 289881 978452 313500 994274 223551 712140 212225 490113 864437 287145 948134 332311 964905 619217 330530 016843 066600 998037 356544 / 5779 127016 748909 094986 667927 666304 897550 201537 672172 684327 799525 226094 950381 356180 131377 653225 244035 788465 702327 464125 > 455 [i]
- residual code [i] would yield OA(455, 104, S4, 39), but
(55, 55+156, 362)-Net in Base 4 — Upper bound on s
There is no (55, 211, 363)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 12 539194 809457 582738 883088 965405 389942 268547 249355 539765 050304 885957 608870 150877 200784 027769 933236 392288 377743 783023 277833 543560 > 4211 [i]