Best Known (39, 88, s)-Nets in Base 2
(39, 88, 33)-Net over F2 — Constructive and digital
Digital (39, 88, 33)-net over F2, using
- net from sequence [i] based on digital (39, 32)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 39 and N(F) ≥ 33, using
(39, 88, 86)-Net over F2 — Upper bound on s (digital)
There is no digital (39, 88, 87)-net over F2, because
- 9 times m-reduction [i] would yield digital (39, 79, 87)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(279, 87, F2, 40) (dual of [87, 8, 41]-code), but
- adding a parity check bit [i] would yield linear OA(280, 88, F2, 41) (dual of [88, 8, 42]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(280, 88, F2, 41) (dual of [88, 8, 42]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(279, 87, F2, 40) (dual of [87, 8, 41]-code), but
(39, 88, 87)-Net in Base 2 — Upper bound on s
There is no (39, 88, 88)-net in base 2, because
- 5 times m-reduction [i] would yield (39, 83, 88)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(283, 88, S2, 44), but
- adding a parity check bit [i] would yield OA(284, 89, S2, 45), but
- the (dual) Plotkin bound shows that M ≥ 464 227514 732017 603087 171584 / 23 > 284 [i]
- adding a parity check bit [i] would yield OA(284, 89, S2, 45), but
- extracting embedded orthogonal array [i] would yield OA(283, 88, S2, 44), but