Best Known (41, 41+17, s)-Nets in Base 32
(41, 41+17, 4140)-Net over F32 — Constructive and digital
Digital (41, 58, 4140)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 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
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- 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
- digital (1, 9, 44)-net over F32, using
(41, 41+17, 8193)-Net in Base 32 — Constructive
(41, 58, 8193)-net in base 32, using
- 321 times duplication [i] based on (40, 57, 8193)-net in base 32, using
- net defined by OOA [i] based on OOA(3257, 8193, S32, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(3257, 65545, S32, 17), using
- 1 times code embedding in larger space [i] based on OA(3256, 65544, S32, 17), using
- discarding parts of the base [i] based on linear OA(25635, 65544, F256, 17) (dual of [65544, 65509, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(25633, 65536, F256, 17) (dual of [65536, 65503, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(25635, 65544, F256, 17) (dual of [65544, 65509, 18]-code), using
- 1 times code embedding in larger space [i] based on OA(3256, 65544, S32, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(3257, 65545, S32, 17), using
- net defined by OOA [i] based on OOA(3257, 8193, S32, 17, 17), using
(41, 41+17, 62719)-Net over F32 — Digital
Digital (41, 58, 62719)-net over F32, using
(41, 41+17, large)-Net in Base 32 — Upper bound on s
There is no (41, 58, large)-net in base 32, because
- 15 times m-reduction [i] would yield (41, 43, large)-net in base 32, but