Best Known (86, 86+98, s)-Nets in Base 2
(86, 86+98, 52)-Net over F2 — Constructive and digital
Digital (86, 184, 52)-net over F2, using
- t-expansion [i] based on digital (85, 184, 52)-net over F2, using
- net from sequence [i] based on digital (85, 51)-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 3 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 (85, 51)-sequence over F2, using
(86, 86+98, 57)-Net over F2 — Digital
Digital (86, 184, 57)-net over F2, using
- t-expansion [i] based on digital (83, 184, 57)-net over F2, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
(86, 86+98, 182)-Net over F2 — Upper bound on s (digital)
There is no digital (86, 184, 183)-net over F2, because
- 10 times m-reduction [i] would yield digital (86, 174, 183)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2174, 183, F2, 88) (dual of [183, 9, 89]-code), but
- residual code [i] would yield linear OA(286, 94, F2, 44) (dual of [94, 8, 45]-code), but
- adding a parity check bit [i] would yield linear OA(287, 95, F2, 45) (dual of [95, 8, 46]-code), but
- “DHM†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(287, 95, F2, 45) (dual of [95, 8, 46]-code), but
- residual code [i] would yield linear OA(286, 94, F2, 44) (dual of [94, 8, 45]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2174, 183, F2, 88) (dual of [183, 9, 89]-code), but
(86, 86+98, 183)-Net in Base 2 — Upper bound on s
There is no (86, 184, 184)-net in base 2, because
- 6 times m-reduction [i] would yield (86, 178, 184)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2178, 184, S2, 92), but
- adding a parity check bit [i] would yield OA(2179, 185, S2, 93), but
- the (dual) Plotkin bound shows that M ≥ 36 779892 980781 332600 600328 651607 420406 849738 932406 452224 / 47 > 2179 [i]
- adding a parity check bit [i] would yield OA(2179, 185, S2, 93), but
- extracting embedded orthogonal array [i] would yield OA(2178, 184, S2, 92), but