Best Known (46, 103, s)-Nets in Base 2
(46, 103, 34)-Net over F2 — Constructive and digital
Digital (46, 103, 34)-net over F2, using
- t-expansion [i] based on digital (45, 103, 34)-net over F2, using
- net from sequence [i] based on digital (45, 33)-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 1 place 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]
- net from sequence [i] based on digital (45, 33)-sequence over F2, using
(46, 103, 101)-Net over F2 — Upper bound on s (digital)
There is no digital (46, 103, 102)-net over F2, because
- 9 times m-reduction [i] would yield digital (46, 94, 102)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(294, 102, F2, 48) (dual of [102, 8, 49]-code), but
- residual code [i] would yield linear OA(246, 53, F2, 24) (dual of [53, 7, 25]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(247, 54, F2, 24) (dual of [54, 7, 25]-code), but
- adding a parity check bit [i] would yield linear OA(248, 55, F2, 25) (dual of [55, 7, 26]-code), but
- “vT4†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(248, 55, F2, 25) (dual of [55, 7, 26]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(247, 54, F2, 24) (dual of [54, 7, 25]-code), but
- residual code [i] would yield linear OA(246, 53, F2, 24) (dual of [53, 7, 25]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(294, 102, F2, 48) (dual of [102, 8, 49]-code), but
(46, 103, 102)-Net in Base 2 — Upper bound on s
There is no (46, 103, 103)-net in base 2, because
- 5 times m-reduction [i] would yield (46, 98, 103)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(298, 103, S2, 52), but
- adding a parity check bit [i] would yield OA(299, 104, S2, 53), but
- the (dual) Plotkin bound shows that M ≥ 20 282409 603651 670423 947251 286016 / 27 > 299 [i]
- adding a parity check bit [i] would yield OA(299, 104, S2, 53), but
- extracting embedded orthogonal array [i] would yield OA(298, 103, S2, 52), but