Best Known (58, 58+143, s)-Nets in Base 4
(58, 58+143, 66)-Net over F4 — Constructive and digital
Digital (58, 201, 66)-net over F4, using
- t-expansion [i] based on digital (49, 201, 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
(58, 58+143, 91)-Net over F4 — Digital
Digital (58, 201, 91)-net over F4, using
- t-expansion [i] based on digital (50, 201, 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
(58, 58+143, 390)-Net over F4 — Upper bound on s (digital)
There is no digital (58, 201, 391)-net over F4, because
- 3 times m-reduction [i] would yield digital (58, 198, 391)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4198, 391, F4, 140) (dual of [391, 193, 141]-code), but
- residual code [i] would yield OA(458, 250, S4, 35), but
- the linear programming bound shows that M ≥ 1 169022 325481 903034 671724 153754 218504 051580 462823 775601 886542 980534 239232 / 13 842163 950449 124319 768889 299435 690739 > 458 [i]
- residual code [i] would yield OA(458, 250, S4, 35), but
- extracting embedded orthogonal array [i] would yield linear OA(4198, 391, F4, 140) (dual of [391, 193, 141]-code), but
(58, 58+143, 395)-Net in Base 4 — Upper bound on s
There is no (58, 201, 396)-net in base 4, because
- 1 times m-reduction [i] would yield (58, 200, 396)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 863325 941171 202327 544824 185899 779564 654436 844518 587998 310721 985634 737740 140058 590124 660795 255223 706609 283455 252083 832044 > 4200 [i]