Best Known (193−137, 193, s)-Nets in Base 4
(193−137, 193, 66)-Net over F4 — Constructive and digital
Digital (56, 193, 66)-net over F4, using
- t-expansion [i] based on digital (49, 193, 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
(193−137, 193, 91)-Net over F4 — Digital
Digital (56, 193, 91)-net over F4, using
- t-expansion [i] based on digital (50, 193, 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
(193−137, 193, 373)-Net over F4 — Upper bound on s (digital)
There is no digital (56, 193, 374)-net over F4, because
- 1 times m-reduction [i] would yield digital (56, 192, 374)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4192, 374, F4, 136) (dual of [374, 182, 137]-code), but
- residual code [i] would yield OA(456, 237, S4, 34), but
- the linear programming bound shows that M ≥ 227 819482 170490 726334 286831 192651 607943 130369 749048 513507 843112 960000 000000 / 43777 691396 586787 052635 611770 906797 710989 > 456 [i]
- residual code [i] would yield OA(456, 237, S4, 34), but
- extracting embedded orthogonal array [i] would yield linear OA(4192, 374, F4, 136) (dual of [374, 182, 137]-code), but
(193−137, 193, 383)-Net in Base 4 — Upper bound on s
There is no (56, 193, 384)-net in base 4, because
- 1 times m-reduction [i] would yield (56, 192, 384)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 43 860403 798467 103649 180914 637083 396299 686470 018177 666335 089597 420813 317989 445493 990337 292757 637029 931056 430254 933431 > 4192 [i]