Best Known (49, ∞, s)-Nets in Base 4
(49, ∞, 66)-Net over F4 — Constructive and digital
Digital (49, m, 66)-net over F4 for arbitrarily large m, 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
(49, ∞, 81)-Net over F4 — Digital
Digital (49, m, 81)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (49, 80)-sequence over F4, using
- t-expansion [i] based on digital (46, 80)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 46 and N(F) ≥ 81, using
- t-expansion [i] based on digital (46, 80)-sequence over F4, using
(49, ∞, 161)-Net in Base 4 — Upper bound on s
There is no (49, m, 162)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (49, 643, 162)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4643, 162, S4, 4, 594), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 076429 979300 970810 649036 055617 303182 745744 882661 311677 785323 875728 142759 963657 615461 617852 021723 896493 142613 001607 558949 816053 955666 606034 659461 700785 571222 584665 170379 070725 041245 714755 399617 124115 355110 724271 034758 206865 308053 878336 729743 184142 632412 126451 114421 497530 757196 889962 307656 999637 013486 386162 231183 813137 605246 665258 991986 717451 178683 533151 595394 019519 274768 513044 522637 197312 / 595 > 4643 [i]
- extracting embedded OOA [i] would yield OOA(4643, 162, S4, 4, 594), but