Best Known (71, 71+179, s)-Nets in Base 4
(71, 71+179, 66)-Net over F4 — Constructive and digital
Digital (71, 250, 66)-net over F4, using
- t-expansion [i] based on digital (49, 250, 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
(71, 71+179, 105)-Net over F4 — Digital
Digital (71, 250, 105)-net over F4, using
- t-expansion [i] based on digital (70, 250, 105)-net over F4, using
- net from sequence [i] based on digital (70, 104)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 70 and N(F) ≥ 105, using
- net from sequence [i] based on digital (70, 104)-sequence over F4, using
(71, 71+179, 429)-Net over F4 — Upper bound on s (digital)
There is no digital (71, 250, 430)-net over F4, because
- 3 times m-reduction [i] would yield digital (71, 247, 430)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4247, 430, F4, 176) (dual of [430, 183, 177]-code), but
- residual code [i] would yield OA(471, 253, S4, 44), but
- the linear programming bound shows that M ≥ 27385 603782 383297 661406 789870 071468 501212 578772 565955 567440 744927 980401 898659 225594 231422 713856 / 4741 255951 754751 821637 567787 914724 985401 817073 767747 > 471 [i]
- residual code [i] would yield OA(471, 253, S4, 44), but
- extracting embedded orthogonal array [i] would yield linear OA(4247, 430, F4, 176) (dual of [430, 183, 177]-code), but
(71, 71+179, 476)-Net in Base 4 — Upper bound on s
There is no (71, 250, 477)-net in base 4, because
- 1 times m-reduction [i] would yield (71, 249, 477)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 863505 704060 429541 804243 069480 506530 558970 684933 069908 094925 683878 562716 964056 571187 899022 254601 916206 651936 279375 925533 817284 760230 748853 620273 238224 > 4249 [i]