Best Known (50−17, 50, s)-Nets in Base 32
(50−17, 50, 4096)-Net over F32 — Constructive and digital
Digital (33, 50, 4096)-net over F32, using
- 321 times duplication [i] based on digital (32, 49, 4096)-net over F32, using
- net defined by OOA [i] based on linear OOA(3249, 4096, F32, 17, 17) (dual of [(4096, 17), 69583, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3249, 32769, F32, 17) (dual of [32769, 32720, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(3249, 32769, F32, 17) (dual of [32769, 32720, 18]-code), using
- net defined by OOA [i] based on linear OOA(3249, 4096, F32, 17, 17) (dual of [(4096, 17), 69583, 18]-NRT-code), using
(50−17, 50, 17102)-Net over F32 — Digital
Digital (33, 50, 17102)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3250, 17102, F32, 17) (dual of [17102, 17052, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3250, 32776, F32, 17) (dual of [32776, 32726, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(3249, 32769, F32, 17) (dual of [32769, 32720, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(3243, 32769, F32, 15) (dual of [32769, 32726, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(321, 7, F32, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3250, 32776, F32, 17) (dual of [32776, 32726, 18]-code), using
(50−17, 50, large)-Net in Base 32 — Upper bound on s
There is no (33, 50, large)-net in base 32, because
- 15 times m-reduction [i] would yield (33, 35, large)-net in base 32, but