Best Known (98−49, 98, s)-Nets in Base 2
(98−49, 98, 35)-Net over F2 — Constructive and digital
Digital (49, 98, 35)-net over F2, using
- t-expansion [i] based on digital (48, 98, 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]
- net from sequence [i] based on digital (48, 34)-sequence over F2, using
(98−49, 98, 36)-Net over F2 — Digital
Digital (49, 98, 36)-net over F2, using
- t-expansion [i] based on digital (47, 98, 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
(98−49, 98, 108)-Net over F2 — Upper bound on s (digital)
There is no digital (49, 98, 109)-net over F2, because
- 1 times m-reduction [i] would yield digital (49, 97, 109)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(297, 109, F2, 48) (dual of [109, 12, 49]-code), but
- construction Y1 [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
- OA(212, 109, S2, 4), but
- discarding factors would yield OA(212, 91, S2, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 4187 > 212 [i]
- discarding factors would yield OA(212, 91, S2, 4), but
- linear OA(296, 105, F2, 48) (dual of [105, 9, 49]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(297, 109, F2, 48) (dual of [109, 12, 49]-code), but
(98−49, 98, 112)-Net in Base 2 — Upper bound on s
There is no (49, 98, 113)-net in base 2, because
- 1 times m-reduction [i] would yield (49, 97, 113)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(297, 113, S2, 48), but
- the linear programming bound shows that M ≥ 12392 552267 831170 629031 770535 755776 / 54375 > 297 [i]
- extracting embedded orthogonal array [i] would yield OA(297, 113, S2, 48), but