Best Known (47−15, 47, s)-Nets in Base 32
(47−15, 47, 4683)-Net over F32 — Constructive and digital
Digital (32, 47, 4683)-net over F32, using
- 321 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(3246, 4683, F32, 15, 15) (dual of [(4683, 15), 70199, 16]-NRT-code), using
(47−15, 47, 9362)-Net in Base 32 — Constructive
(32, 47, 9362)-net in base 32, using
- net defined by OOA [i] based on OOA(3247, 9362, S32, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(3247, 65535, S32, 15), using
- discarding factors based on OA(3247, 65538, S32, 15), using
- discarding parts of the base [i] based on linear OA(25629, 65538, F256, 15) (dual of [65538, 65509, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(25629, 65536, F256, 15) (dual of [65536, 65507, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(25629, 65538, F256, 15) (dual of [65538, 65509, 16]-code), using
- discarding factors based on OA(3247, 65538, S32, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(3247, 65535, S32, 15), using
(47−15, 47, 32787)-Net over F32 — Digital
Digital (32, 47, 32787)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3247, 32787, F32, 15) (dual of [32787, 32740, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- linear OA(3243, 32768, F32, 15) (dual of [32768, 32725, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(3228, 32768, F32, 10) (dual of [32768, 32740, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(324, 19, F32, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
(47−15, 47, large)-Net in Base 32 — Upper bound on s
There is no (32, 47, large)-net in base 32, because
- 13 times m-reduction [i] would yield (32, 34, large)-net in base 32, but