Best Known (66−21, 66, s)-Nets in Base 32
(66−21, 66, 3279)-Net over F32 — Constructive and digital
Digital (45, 66, 3279)-net over F32, using
- net defined by OOA [i] based on linear OOA(3266, 3279, F32, 21, 21) (dual of [(3279, 21), 68793, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3266, 32791, F32, 21) (dual of [32791, 32725, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3266, 32792, F32, 21) (dual of [32792, 32726, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- 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(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(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([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3266, 32792, F32, 21) (dual of [32792, 32726, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3266, 32791, F32, 21) (dual of [32791, 32725, 22]-code), using
(66−21, 66, 6553)-Net in Base 32 — Constructive
(45, 66, 6553)-net in base 32, using
- net defined by OOA [i] based on OOA(3266, 6553, S32, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(3266, 65531, S32, 21), using
- discarding factors based on OA(3266, 65538, S32, 21), using
- discarding parts of the base [i] based on linear OA(25641, 65538, F256, 21) (dual of [65538, 65497, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- 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(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(20) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(25641, 65538, F256, 21) (dual of [65538, 65497, 22]-code), using
- discarding factors based on OA(3266, 65538, S32, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(3266, 65531, S32, 21), using
(66−21, 66, 32792)-Net over F32 — Digital
Digital (45, 66, 32792)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3266, 32792, F32, 21) (dual of [32792, 32726, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- 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(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(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([0,10]) ⊂ C([0,7]) [i] based on
(66−21, 66, large)-Net in Base 32 — Upper bound on s
There is no (45, 66, large)-net in base 32, because
- 19 times m-reduction [i] would yield (45, 47, large)-net in base 32, but