Best Known (42−41, 42, s)-Nets in Base 25
(42−41, 42, 27)-Net over F25 — Constructive and digital
Digital (1, 42, 27)-net over F25, using
- net from sequence [i] based on digital (1, 26)-sequence over F25, using
(42−41, 42, 36)-Net over F25 — Digital
Digital (1, 42, 36)-net over F25, using
- net from sequence [i] based on digital (1, 35)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 1 and N(F) ≥ 36, using
(42−41, 42, 74)-Net over F25 — Upper bound on s (digital)
There is no digital (1, 42, 75)-net over F25, because
- 16 times m-reduction [i] would yield digital (1, 26, 75)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(2526, 75, F25, 25) (dual of [75, 49, 26]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(2549, 75, F25, 48) (dual of [75, 26, 49]-code), but
- discarding factors / shortening the dual code would yield linear OA(2549, 52, F25, 48) (dual of [52, 3, 49]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(2549, 75, F25, 48) (dual of [75, 26, 49]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2526, 75, F25, 25) (dual of [75, 49, 26]-code), but
(42−41, 42, 93)-Net in Base 25 — Upper bound on s
There is no (1, 42, 94)-net in base 25, because
- 4 times m-reduction [i] would yield (1, 38, 94)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(2538, 94, S25, 37), but
- the linear programming bound shows that M ≥ 789627 539185 406277 823242 593427 721658 372320 234775 543212 890625 / 5 958181 > 2538 [i]
- extracting embedded orthogonal array [i] would yield OA(2538, 94, S25, 37), but