Best Known (234−173, 234, s)-Nets in Base 4
(234−173, 234, 66)-Net over F4 — Constructive and digital
Digital (61, 234, 66)-net over F4, using
- t-expansion [i] based on digital (49, 234, 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
(234−173, 234, 99)-Net over F4 — Digital
Digital (61, 234, 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
(234−173, 234, 287)-Net over F4 — Upper bound on s (digital)
There is no digital (61, 234, 288)-net over F4, because
- 1 times m-reduction [i] would yield digital (61, 233, 288)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4233, 288, F4, 172) (dual of [288, 55, 173]-code), but
- residual code [i] would yield OA(461, 115, S4, 43), but
- the linear programming bound shows that M ≥ 8 867577 820394 892078 302903 376011 219649 893047 268294 051459 959774 715216 956970 670840 325873 158120 379501 398960 032970 458008 737468 823847 270763 370822 590292 322808 758272 / 1 568698 363933 661749 730985 011129 979312 182529 988673 402786 941022 163461 443141 672915 973490 090934 276533 827022 911521 994930 215625 > 461 [i]
- residual code [i] would yield OA(461, 115, S4, 43), but
- extracting embedded orthogonal array [i] would yield linear OA(4233, 288, F4, 172) (dual of [288, 55, 173]-code), but
(234−173, 234, 400)-Net in Base 4 — Upper bound on s
There is no (61, 234, 401)-net in base 4, because
- 1 times m-reduction [i] would yield (61, 233, 401)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 207 289683 636672 296379 057877 553675 983084 747370 630303 382241 581408 275114 444994 638467 165112 268732 032808 556617 475391 958006 009707 390771 939636 517952 > 4233 [i]