Best Known (240−170, 240, s)-Nets in Base 4
(240−170, 240, 66)-Net over F4 — Constructive and digital
Digital (70, 240, 66)-net over F4, using
- t-expansion [i] based on digital (49, 240, 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
(240−170, 240, 105)-Net over F4 — Digital
Digital (70, 240, 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
(240−170, 240, 453)-Net over F4 — Upper bound on s (digital)
There is no digital (70, 240, 454)-net over F4, because
- 2 times m-reduction [i] would yield digital (70, 238, 454)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4238, 454, F4, 168) (dual of [454, 216, 169]-code), but
- residual code [i] would yield OA(470, 285, S4, 42), but
- the linear programming bound shows that M ≥ 2 520198 658998 621179 007148 715142 691743 667539 102383 827900 254205 426285 806015 463738 326023 123475 890176 / 1 791905 782922 287300 464200 415513 167226 158849 588749 853743 > 470 [i]
- residual code [i] would yield OA(470, 285, S4, 42), but
- extracting embedded orthogonal array [i] would yield linear OA(4238, 454, F4, 168) (dual of [454, 216, 169]-code), but
(240−170, 240, 474)-Net in Base 4 — Upper bound on s
There is no (70, 240, 475)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 3 165110 250434 729359 649171 389047 468488 437767 851533 222814 483110 505694 097074 933010 661026 395132 897878 231420 809825 667141 333772 090202 550435 452732 700572 > 4240 [i]