Best Known (84−27, 84, s)-Nets in Base 32
(84−27, 84, 2522)-Net over F32 — Constructive and digital
Digital (57, 84, 2522)-net over F32, using
- 321 times duplication [i] based on digital (56, 83, 2522)-net over F32, using
- net defined by OOA [i] based on linear OOA(3283, 2522, F32, 27, 27) (dual of [(2522, 27), 68011, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3283, 32787, F32, 27) (dual of [32787, 32704, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- linear OA(3279, 32768, F32, 27) (dual of [32768, 32689, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- 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(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(26) ⊂ Ce(21) [i] based on
- OOA 13-folding and stacking with additional row [i] based on linear OA(3283, 32787, F32, 27) (dual of [32787, 32704, 28]-code), using
- net defined by OOA [i] based on linear OOA(3283, 2522, F32, 27, 27) (dual of [(2522, 27), 68011, 28]-NRT-code), using
(84−27, 84, 32599)-Net over F32 — Digital
Digital (57, 84, 32599)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3284, 32599, F32, 27) (dual of [32599, 32515, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3284, 32792, F32, 27) (dual of [32792, 32708, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(3279, 32769, F32, 27) (dual of [32769, 32690, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3261, 32769, F32, 21) (dual of [32769, 32708, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [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 C([0,13]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3284, 32792, F32, 27) (dual of [32792, 32708, 28]-code), using
(84−27, 84, large)-Net in Base 32 — Upper bound on s
There is no (57, 84, large)-net in base 32, because
- 25 times m-reduction [i] would yield (57, 59, large)-net in base 32, but