Best Known (230−170, 230, s)-Nets in Base 4
(230−170, 230, 66)-Net over F4 — Constructive and digital
Digital (60, 230, 66)-net over F4, using
- t-expansion [i] based on digital (49, 230, 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
(230−170, 230, 91)-Net over F4 — Digital
Digital (60, 230, 91)-net over F4, using
- t-expansion [i] based on digital (50, 230, 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
(230−170, 230, 288)-Net over F4 — Upper bound on s (digital)
There is no digital (60, 230, 289)-net over F4, because
- 2 times m-reduction [i] would yield digital (60, 228, 289)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4228, 289, F4, 168) (dual of [289, 61, 169]-code), but
- residual code [i] would yield OA(460, 120, S4, 42), but
- the linear programming bound shows that M ≥ 290 323585 871668 540958 706283 871420 089371 226973 685105 944092 049613 090277 543017 758174 630279 499647 453693 213959 295987 570365 373495 501567 458190 834202 690982 141368 167162 106346 363122 321383 704967 027230 420318 677240 681705 674259 718795 341724 668084 100666 837764 012913 983488 / 217 009749 131050 124974 338662 159260 162741 593420 668312 475869 644870 983601 362036 477462 521260 959668 929843 611506 134957 195877 747489 996242 167742 452236 130306 867798 356780 551051 603934 664707 901784 404757 406481 175288 812407 108150 526731 > 460 [i]
- residual code [i] would yield OA(460, 120, S4, 42), but
- extracting embedded orthogonal array [i] would yield linear OA(4228, 289, F4, 168) (dual of [289, 61, 169]-code), but
(230−170, 230, 393)-Net in Base 4 — Upper bound on s
There is no (60, 230, 394)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 3 040116 840087 530889 823205 088360 483750 843039 003920 018321 297875 003499 413577 359587 499740 055148 611803 637972 756442 379404 091290 735288 269376 053360 > 4230 [i]