Best Known (48, 48+57, s)-Nets in Base 2
(48, 48+57, 35)-Net over F2 — Constructive and digital
Digital (48, 105, 35)-net over F2, using
- net from sequence [i] based on digital (48, 34)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 41, N(F) = 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F2 with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
(48, 48+57, 36)-Net over F2 — Digital
Digital (48, 105, 36)-net over F2, using
- t-expansion [i] based on digital (47, 105, 36)-net over F2, using
- net from sequence [i] based on digital (47, 35)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 47 and N(F) ≥ 36, using
- net from sequence [i] based on digital (47, 35)-sequence over F2, using
(48, 48+57, 104)-Net over F2 — Upper bound on s (digital)
There is no digital (48, 105, 105)-net over F2, because
- 9 times m-reduction [i] would yield digital (48, 96, 105)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(296, 105, F2, 48) (dual of [105, 9, 49]-code), but
- residual code [i] would yield linear OA(248, 56, F2, 24) (dual of [56, 8, 25]-code), but
- adding a parity check bit [i] would yield linear OA(249, 57, F2, 25) (dual of [57, 8, 26]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(249, 57, F2, 25) (dual of [57, 8, 26]-code), but
- residual code [i] would yield linear OA(248, 56, F2, 24) (dual of [56, 8, 25]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(296, 105, F2, 48) (dual of [105, 9, 49]-code), but
(48, 48+57, 106)-Net in Base 2 — Upper bound on s
There is no (48, 105, 107)-net in base 2, because
- 3 times m-reduction [i] would yield (48, 102, 107)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2102, 107, S2, 54), but
- adding a parity check bit [i] would yield OA(2103, 108, S2, 55), but
- the (dual) Plotkin bound shows that M ≥ 81 129638 414606 681695 789005 144064 / 7 > 2103 [i]
- adding a parity check bit [i] would yield OA(2103, 108, S2, 55), but
- extracting embedded orthogonal array [i] would yield OA(2102, 107, S2, 54), but