Best Known (48, 48+22, s)-Nets in Base 32
(48, 48+22, 2981)-Net over F32 — Constructive and digital
Digital (48, 70, 2981)-net over F32, using
- 321 times duplication [i] based on digital (47, 69, 2981)-net over F32, using
- net defined by OOA [i] based on linear OOA(3269, 2981, F32, 22, 22) (dual of [(2981, 22), 65513, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3269, 32791, F32, 22) (dual of [32791, 32722, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(3264, 32768, F32, 22) (dual of [32768, 32704, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3246, 32768, F32, 16) (dual of [32768, 32722, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(325, 23, F32, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- OA 11-folding and stacking [i] based on linear OA(3269, 32791, F32, 22) (dual of [32791, 32722, 23]-code), using
- net defined by OOA [i] based on linear OOA(3269, 2981, F32, 22, 22) (dual of [(2981, 22), 65513, 23]-NRT-code), using
(48, 48+22, 5958)-Net in Base 32 — Constructive
(48, 70, 5958)-net in base 32, using
- 321 times duplication [i] based on (47, 69, 5958)-net in base 32, using
- net defined by OOA [i] based on OOA(3269, 5958, S32, 22, 22), using
- OA 11-folding and stacking [i] based on OA(3269, 65538, S32, 22), using
- discarding parts of the base [i] based on linear OA(25643, 65538, F256, 22) (dual of [65538, 65495, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(25643, 65536, F256, 22) (dual of [65536, 65493, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(25641, 65536, F256, 21) (dual of [65536, 65495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- discarding parts of the base [i] based on linear OA(25643, 65538, F256, 22) (dual of [65538, 65495, 23]-code), using
- OA 11-folding and stacking [i] based on OA(3269, 65538, S32, 22), using
- net defined by OOA [i] based on OOA(3269, 5958, S32, 22, 22), using
(48, 48+22, 32795)-Net over F32 — Digital
Digital (48, 70, 32795)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3270, 32795, F32, 22) (dual of [32795, 32725, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(14) [i] based on
- linear OA(3264, 32768, F32, 22) (dual of [32768, 32704, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(326, 27, F32, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,32)), using
- discarding factors / shortening the dual code based on linear OA(326, 32, F32, 6) (dual of [32, 26, 7]-code or 32-arc in PG(5,32)), using
- Reed–Solomon code RS(26,32) [i]
- discarding factors / shortening the dual code based on linear OA(326, 32, F32, 6) (dual of [32, 26, 7]-code or 32-arc in PG(5,32)), using
- construction X applied to Ce(21) ⊂ Ce(14) [i] based on
(48, 48+22, large)-Net in Base 32 — Upper bound on s
There is no (48, 70, large)-net in base 32, because
- 20 times m-reduction [i] would yield (48, 50, large)-net in base 32, but