Best Known (202−96, 202, s)-Nets in Base 2
(202−96, 202, 56)-Net over F2 — Constructive and digital
Digital (106, 202, 56)-net over F2, using
- t-expansion [i] based on digital (105, 202, 56)-net over F2, using
- net from sequence [i] based on digital (105, 55)-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 7 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 (105, 55)-sequence over F2, using
(202−96, 202, 65)-Net over F2 — Digital
Digital (106, 202, 65)-net over F2, using
- t-expansion [i] based on digital (95, 202, 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
(202−96, 202, 263)-Net over F2 — Upper bound on s (digital)
There is no digital (106, 202, 264)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2202, 264, F2, 96) (dual of [264, 62, 97]-code), but
- construction Y1 [i] would yield
- linear OA(2201, 242, F2, 96) (dual of [242, 41, 97]-code), but
- residual code [i] would yield OA(2105, 145, S2, 48), but
- the linear programming bound shows that M ≥ 727 070369 678229 731686 401086 929362 121333 407744 / 17 521374 765895 > 2105 [i]
- residual code [i] would yield OA(2105, 145, S2, 48), but
- OA(262, 264, S2, 22), but
- discarding factors would yield OA(262, 249, S2, 22), but
- the Rao or (dual) Hamming bound shows that M ≥ 4 787396 297110 060300 > 262 [i]
- discarding factors would yield OA(262, 249, S2, 22), but
- linear OA(2201, 242, F2, 96) (dual of [242, 41, 97]-code), but
- construction Y1 [i] would yield
(202−96, 202, 281)-Net in Base 2 — Upper bound on s
There is no (106, 202, 282)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 7 389022 085454 291760 997068 036326 968292 590399 495691 474282 381720 > 2202 [i]
- extracting embedded orthogonal array [i] would yield OA(2202, 282, S2, 96), but
- 17 times code embedding in larger space [i] would yield OA(2219, 299, S2, 96), but
- adding a parity check bit [i] would yield OA(2220, 300, S2, 97), but
- the linear programming bound shows that M ≥ 2 707506 179671 220537 436761 186277 555172 982275 381427 765331 317661 637213 928123 767152 572244 071765 637582 553088 / 1 434637 915544 533146 412054 075529 090625 > 2220 [i]
- adding a parity check bit [i] would yield OA(2220, 300, S2, 97), but
- 17 times code embedding in larger space [i] would yield OA(2219, 299, S2, 96), but