Best Known (126, 126+∞, s)-Nets in Base 2
(126, 126+∞, 57)-Net over F2 — Constructive and digital
Digital (126, m, 57)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (126, 56)-sequence over F2, using
- t-expansion [i] based on digital (110, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 8 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- t-expansion [i] based on digital (110, 56)-sequence over F2, using
(126, 126+∞, 81)-Net over F2 — Digital
Digital (126, m, 81)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (126, 80)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 126 and N(F) ≥ 81, using
(126, 126+∞, 136)-Net in Base 2 — Upper bound on s
There is no (126, m, 137)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (126, 949, 137)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2949, 137, S2, 7, 823), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 566256 038748 339790 313502 199295 482133 297983 884791 897410 008437 924027 510202 246681 985446 000094 160136 338304 595354 019305 484137 613106 061503 500093 526748 285283 791873 498155 296042 392719 997374 984864 900578 200187 085753 919427 288070 598707 579096 737570 223276 484768 998010 986776 156938 002003 289844 828631 728128 / 103 > 2949 [i]
- extracting embedded OOA [i] would yield OOA(2949, 137, S2, 7, 823), but