Best Known (215−158, 215, s)-Nets in Base 4
(215−158, 215, 66)-Net over F4 — Constructive and digital
Digital (57, 215, 66)-net over F4, using
- t-expansion [i] based on digital (49, 215, 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
(215−158, 215, 91)-Net over F4 — Digital
Digital (57, 215, 91)-net over F4, using
- t-expansion [i] based on digital (50, 215, 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
(215−158, 215, 290)-Net over F4 — Upper bound on s (digital)
There is no digital (57, 215, 291)-net over F4, because
- 2 times m-reduction [i] would yield digital (57, 213, 291)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4213, 291, F4, 156) (dual of [291, 78, 157]-code), but
- residual code [i] would yield OA(457, 134, S4, 39), but
- the linear programming bound shows that M ≥ 2118 872152 947773 735205 614860 840104 993723 173522 547123 836257 668222 520389 144284 698869 497944 229365 678080 000000 / 96823 132929 101376 060431 157559 643789 049324 330608 249156 649224 680559 609039 > 457 [i]
- residual code [i] would yield OA(457, 134, S4, 39), but
- extracting embedded orthogonal array [i] would yield linear OA(4213, 291, F4, 156) (dual of [291, 78, 157]-code), but
(215−158, 215, 376)-Net in Base 4 — Upper bound on s
There is no (57, 215, 377)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 3067 873263 161998 151136 665320 046258 579378 529213 930571 145912 758446 451858 117740 092310 425208 733936 067237 521951 283889 380102 810076 708016 > 4215 [i]