Best Known (88, 88+105, s)-Nets in Base 2
(88, 88+105, 52)-Net over F2 — Constructive and digital
Digital (88, 193, 52)-net over F2, using
- t-expansion [i] based on digital (85, 193, 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
(88, 88+105, 57)-Net over F2 — Digital
Digital (88, 193, 57)-net over F2, using
- t-expansion [i] based on digital (83, 193, 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
(88, 88+105, 186)-Net over F2 — Upper bound on s (digital)
There is no digital (88, 193, 187)-net over F2, because
- 17 times m-reduction [i] would yield digital (88, 176, 187)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2176, 187, F2, 88) (dual of [187, 11, 89]-code), but
- residual code [i] would yield linear OA(288, 98, F2, 44) (dual of [98, 10, 45]-code), but
- adding a parity check bit [i] would yield linear OA(289, 99, F2, 45) (dual of [99, 10, 46]-code), but
- residual code [i] would yield linear OA(288, 98, F2, 44) (dual of [98, 10, 45]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2176, 187, F2, 88) (dual of [187, 11, 89]-code), but
(88, 88+105, 188)-Net in Base 2 — Upper bound on s
There is no (88, 193, 189)-net in base 2, because
- 11 times m-reduction [i] would yield (88, 182, 189)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2182, 189, S2, 94), but
- adding a parity check bit [i] would yield OA(2183, 190, S2, 95), but
- the (dual) Plotkin bound shows that M ≥ 49 039857 307708 443467 467104 868809 893875 799651 909875 269632 / 3 > 2183 [i]
- adding a parity check bit [i] would yield OA(2183, 190, S2, 95), but
- extracting embedded orthogonal array [i] would yield OA(2182, 189, S2, 94), but