Best Known (70, 70+175, s)-Nets in Base 4
(70, 70+175, 66)-Net over F4 — Constructive and digital
Digital (70, 245, 66)-net over F4, using
- t-expansion [i] based on digital (49, 245, 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
(70, 70+175, 105)-Net over F4 — Digital
Digital (70, 245, 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
(70, 70+175, 434)-Net over F4 — Upper bound on s (digital)
There is no digital (70, 245, 435)-net over F4, because
- 3 times m-reduction [i] would yield digital (70, 242, 435)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4242, 435, F4, 172) (dual of [435, 193, 173]-code), but
- residual code [i] would yield OA(470, 262, S4, 43), but
- the linear programming bound shows that M ≥ 45 983165 517068 297448 511524 093141 662826 825837 816964 700852 432473 444189 329902 728330 463517 081600 / 32 437855 549903 223644 246089 638836 795109 698224 516577 > 470 [i]
- residual code [i] would yield OA(470, 262, S4, 43), but
- extracting embedded orthogonal array [i] would yield linear OA(4242, 435, F4, 172) (dual of [435, 193, 173]-code), but
(70, 70+175, 471)-Net in Base 4 — Upper bound on s
There is no (70, 245, 472)-net in base 4, because
- 1 times m-reduction [i] would yield (70, 244, 472)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 880 043440 910925 217062 980358 139075 992018 800851 320455 738641 760842 617624 394436 768788 233210 215033 854905 636588 907698 806869 945828 858750 558474 149040 501645 > 4244 [i]