Best Known (1, 1+34, s)-Nets in Base 32
(1, 1+34, 44)-Net over F32 — Constructive and digital
Digital (1, 35, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
(1, 1+34, 95)-Net over F32 — Upper bound on s (digital)
There is no digital (1, 35, 96)-net over F32, because
- 2 times m-reduction [i] would yield digital (1, 33, 96)-net over F32, but
- extracting embedded orthogonal array [i] would yield linear OA(3233, 96, F32, 32) (dual of [96, 63, 33]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(3263, 96, F32, 62) (dual of [96, 33, 63]-code), but
- discarding factors / shortening the dual code would yield linear OA(3263, 66, F32, 62) (dual of [66, 3, 63]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(3263, 96, F32, 62) (dual of [96, 33, 63]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(3233, 96, F32, 32) (dual of [96, 63, 33]-code), but
(1, 1+34, 165)-Net in Base 32 — Upper bound on s
There is no (1, 35, 166)-net in base 32, because
- 26 times m-reduction [i] would yield (1, 9, 166)-net in base 32, but
- extracting embedded orthogonal array [i] would yield OA(329, 166, S32, 8), but
- the linear programming bound shows that M ≥ 2 187320 464516 698429 456384 / 61939 225499 > 329 [i]
- extracting embedded orthogonal array [i] would yield OA(329, 166, S32, 8), but