Best Known (31, 31+15, s)-Nets in Base 32
(31, 31+15, 4683)-Net over F32 — Constructive and digital
Digital (31, 46, 4683)-net over F32, using
- net defined by OOA [i] based on linear OOA(3246, 4683, F32, 15, 15) (dual of [(4683, 15), 70199, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3246, 32782, F32, 15) (dual of [32782, 32736, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3246, 32784, F32, 15) (dual of [32784, 32738, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- 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(3231, 32769, F32, 11) (dual of [32769, 32738, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(323, 15, F32, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,32) or 15-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([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3246, 32784, F32, 15) (dual of [32784, 32738, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3246, 32782, F32, 15) (dual of [32782, 32736, 16]-code), using
(31, 31+15, 29655)-Net over F32 — Digital
Digital (31, 46, 29655)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3246, 29655, F32, 15) (dual of [29655, 29609, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3246, 32784, F32, 15) (dual of [32784, 32738, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- 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(3231, 32769, F32, 11) (dual of [32769, 32738, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(323, 15, F32, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,32) or 15-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([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3246, 32784, F32, 15) (dual of [32784, 32738, 16]-code), using
(31, 31+15, large)-Net in Base 32 — Upper bound on s
There is no (31, 46, large)-net in base 32, because
- 13 times m-reduction [i] would yield (31, 33, large)-net in base 32, but