Best Known (13, 24, s)-Nets in Base 32
(13, 24, 207)-Net over F32 — Constructive and digital
Digital (13, 24, 207)-net over F32, using
- net defined by OOA [i] based on linear OOA(3224, 207, F32, 11, 11) (dual of [(207, 11), 2253, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3224, 1036, F32, 11) (dual of [1036, 1012, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- linear OA(3221, 1025, F32, 11) (dual of [1025, 1004, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(3213, 1025, F32, 7) (dual of [1025, 1012, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,5]) ⊂ C([0,3]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(3224, 1036, F32, 11) (dual of [1036, 1012, 12]-code), using
(13, 24, 322)-Net in Base 32 — Constructive
(13, 24, 322)-net in base 32, using
- (u, u+v)-construction [i] based on
- (1, 6, 65)-net in base 32, using
- base change [i] based on digital (0, 5, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- base change [i] based on digital (0, 5, 65)-net over F64, using
- (7, 18, 257)-net in base 32, using
- base change [i] based on (4, 15, 257)-net in base 64, using
- 1 times m-reduction [i] based on (4, 16, 257)-net in base 64, using
- base change [i] based on digital (0, 12, 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, 12, 257)-net over F256, using
- 1 times m-reduction [i] based on (4, 16, 257)-net in base 64, using
- base change [i] based on (4, 15, 257)-net in base 64, using
- (1, 6, 65)-net in base 32, using
(13, 24, 936)-Net over F32 — Digital
Digital (13, 24, 936)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3224, 936, F32, 11) (dual of [936, 912, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3224, 1036, F32, 11) (dual of [1036, 1012, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- linear OA(3221, 1025, F32, 11) (dual of [1025, 1004, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(3213, 1025, F32, 7) (dual of [1025, 1012, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,5]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3224, 1036, F32, 11) (dual of [1036, 1012, 12]-code), using
(13, 24, 704957)-Net in Base 32 — Upper bound on s
There is no (13, 24, 704958)-net in base 32, because
- 1 times m-reduction [i] would yield (13, 23, 704958)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 41538 431866 649007 149744 730382 932933 > 3223 [i]