Best Known (64, 238, s)-Nets in Base 4
(64, 238, 66)-Net over F4 — Constructive and digital
Digital (64, 238, 66)-net over F4, using
- t-expansion [i] based on digital (49, 238, 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
(64, 238, 99)-Net over F4 — Digital
Digital (64, 238, 99)-net over F4, using
- t-expansion [i] based on digital (61, 238, 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
(64, 238, 331)-Net over F4 — Upper bound on s (digital)
There is no digital (64, 238, 332)-net over F4, because
- 2 times m-reduction [i] would yield digital (64, 236, 332)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4236, 332, F4, 172) (dual of [332, 96, 173]-code), but
- residual code [i] would yield OA(464, 159, S4, 43), but
- the linear programming bound shows that M ≥ 45270 819771 373261 134814 974786 535179 781633 219686 501979 035125 061347 877742 643588 123815 870907 832622 000802 701749 559691 837440 / 132 156573 038196 016753 662455 730323 092222 536469 928086 508825 109263 446128 015973 837189 > 464 [i]
- residual code [i] would yield OA(464, 159, S4, 43), but
- extracting embedded orthogonal array [i] would yield linear OA(4236, 332, F4, 172) (dual of [332, 96, 173]-code), but
(64, 238, 422)-Net in Base 4 — Upper bound on s
There is no (64, 238, 423)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 213676 648975 099314 455058 874434 020465 671437 537072 051023 068390 956430 117238 324783 558209 287884 974442 803269 172205 976215 567546 629852 914164 949970 572304 > 4238 [i]