Best Known (52−51, 52, s)-Nets in Base 49
(52−51, 52, 51)-Net over F49 — Constructive and digital
Digital (1, 52, 51)-net over F49, using
- net from sequence [i] based on digital (1, 50)-sequence over F49, using
(52−51, 52, 64)-Net over F49 — Digital
Digital (1, 52, 64)-net over F49, using
- net from sequence [i] based on digital (1, 63)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 1 and N(F) ≥ 64, using
(52−51, 52, 146)-Net over F49 — Upper bound on s (digital)
There is no digital (1, 52, 147)-net over F49, because
- 2 times m-reduction [i] would yield digital (1, 50, 147)-net over F49, but
- extracting embedded orthogonal array [i] would yield linear OA(4950, 147, F49, 49) (dual of [147, 97, 50]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(4997, 147, F49, 96) (dual of [147, 50, 97]-code), but
- discarding factors / shortening the dual code would yield linear OA(4997, 100, F49, 96) (dual of [100, 3, 97]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(4997, 147, F49, 96) (dual of [147, 50, 97]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(4950, 147, F49, 49) (dual of [147, 97, 50]-code), but
(52−51, 52, 281)-Net in Base 49 — Upper bound on s
There is no (1, 52, 282)-net in base 49, because
- 41 times m-reduction [i] would yield (1, 11, 282)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 3 929889 617545 981537 > 4911 [i]