Best Known (27, 27+25, s)-Nets in Base 32
(27, 27+25, 196)-Net over F32 — Constructive and digital
Digital (27, 52, 196)-net over F32, using
- 1 times m-reduction [i] based on digital (27, 53, 196)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (7, 33, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32 (see above)
- digital (7, 20, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(27, 27+25, 290)-Net in Base 32 — Constructive
(27, 52, 290)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (0, 12, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- (15, 40, 257)-net in base 32, using
- base change [i] based on digital (0, 25, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 25, 257)-net over F256, using
- digital (0, 12, 33)-net over F32, using
(27, 27+25, 651)-Net over F32 — Digital
Digital (27, 52, 651)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3252, 651, F32, 25) (dual of [651, 599, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3252, 1036, F32, 25) (dual of [1036, 984, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(3249, 1025, F32, 25) (dual of [1025, 976, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3241, 1025, F32, 21) (dual of [1025, 984, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(323, 11, F32, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,32) or 11-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3252, 1036, F32, 25) (dual of [1036, 984, 26]-code), using
(27, 27+25, 425481)-Net in Base 32 — Upper bound on s
There is no (27, 52, 425482)-net in base 32, because
- 1 times m-reduction [i] would yield (27, 51, 425482)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 57896 161825 025783 566625 456999 569538 819363 583537 617640 510267 130786 630590 474624 > 3251 [i]