Best Known (208−151, 208, s)-Nets in Base 4
(208−151, 208, 66)-Net over F4 — Constructive and digital
Digital (57, 208, 66)-net over F4, using
- t-expansion [i] based on digital (49, 208, 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
(208−151, 208, 91)-Net over F4 — Digital
Digital (57, 208, 91)-net over F4, using
- t-expansion [i] based on digital (50, 208, 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
(208−151, 208, 325)-Net over F4 — Upper bound on s (digital)
There is no digital (57, 208, 326)-net over F4, because
- 3 times m-reduction [i] would yield digital (57, 205, 326)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4205, 326, F4, 148) (dual of [326, 121, 149]-code), but
- residual code [i] would yield OA(457, 177, S4, 37), but
- the linear programming bound shows that M ≥ 7 916017 728459 289141 982805 636437 938856 538720 259539 351050 169416 698004 574623 120920 412160 / 363 505843 153703 749393 199977 171265 253621 464016 748729 > 457 [i]
- residual code [i] would yield OA(457, 177, S4, 37), but
- extracting embedded orthogonal array [i] would yield linear OA(4205, 326, F4, 148) (dual of [326, 121, 149]-code), but
(208−151, 208, 380)-Net in Base 4 — Upper bound on s
There is no (57, 208, 381)-net in base 4, because
- 1 times m-reduction [i] would yield (57, 207, 381)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 43692 411990 686562 954214 934273 060902 937668 658533 332212 465306 579567 134135 140051 966249 482931 820155 008911 126541 518828 089360 317404 > 4207 [i]