Best Known (27, ∞, s)-Nets in Base 9
(27, ∞, 78)-Net over F9 — Constructive and digital
Digital (27, m, 78)-net over F9 for arbitrarily large m, using
- net from sequence [i] based on digital (27, 77)-sequence over F9, using
- t-expansion [i] based on digital (22, 77)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 22 and N(F) ≥ 78, using
- F3 from the tower of function fields by GarcÃa and Stichtenoth over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 22 and N(F) ≥ 78, using
- t-expansion [i] based on digital (22, 77)-sequence over F9, using
(27, ∞, 110)-Net over F9 — Digital
Digital (27, m, 110)-net over F9 for arbitrarily large m, using
- net from sequence [i] based on digital (27, 109)-sequence over F9, using
- t-expansion [i] based on digital (26, 109)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 26 and N(F) ≥ 110, using
- t-expansion [i] based on digital (26, 109)-sequence over F9, using
(27, ∞, 239)-Net in Base 9 — Upper bound on s
There is no (27, m, 240)-net in base 9 for arbitrarily large m, because
- m-reduction [i] would yield (27, 477, 240)-net in base 9, but
- extracting embedded OOA [i] would yield OOA(9477, 240, S9, 2, 450), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 68 468322 575345 128708 991945 977526 175517 118999 790686 889404 205139 759723 787216 005853 509496 112819 740635 720215 209225 685549 104067 668360 355149 897401 505089 666738 172567 067159 547581 988947 639005 597437 010108 899055 392848 725631 659971 645504 595683 039930 824551 684834 154579 903072 244781 080018 706633 077188 797002 430775 666344 165994 040622 187758 532352 490821 994793 584635 327917 894072 654094 148502 948302 696176 369303 307965 159160 698929 324011 699554 201081 757304 858875 332084 387238 629274 645571 / 451 > 9477 [i]
- extracting embedded OOA [i] would yield OOA(9477, 240, S9, 2, 450), but