Best Known (182−83, 182, s)-Nets in Base 2
(182−83, 182, 54)-Net over F2 — Constructive and digital
Digital (99, 182, 54)-net over F2, using
- t-expansion [i] based on digital (95, 182, 54)-net over F2, using
- net from sequence [i] based on digital (95, 53)-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 5 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]
- net from sequence [i] based on digital (95, 53)-sequence over F2, using
(182−83, 182, 65)-Net over F2 — Digital
Digital (99, 182, 65)-net over F2, using
- t-expansion [i] based on digital (95, 182, 65)-net over F2, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 95 and N(F) ≥ 65, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
(182−83, 182, 287)-Net in Base 2 — Upper bound on s
There is no (99, 182, 288)-net in base 2, because
- 1 times m-reduction [i] would yield (99, 181, 288)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 162654 387371 884515 720854 925241 290817 532490 695337 593256 > 2181 [i]
- extracting embedded orthogonal array [i] would yield OA(2181, 288, S2, 82), but
- 11 times code embedding in larger space [i] would yield OA(2192, 299, S2, 82), but
- adding a parity check bit [i] would yield OA(2193, 300, S2, 83), but
- the linear programming bound shows that M ≥ 47301 037877 562348 126535 964476 521603 387556 508620 623293 070152 196591 060102 430290 310541 234357 121691 993687 546336 928736 477184 / 2 739614 630568 907331 852802 953012 407724 450650 903550 704398 106875 > 2193 [i]
- adding a parity check bit [i] would yield OA(2193, 300, S2, 83), but
- 11 times code embedding in larger space [i] would yield OA(2192, 299, S2, 82), but