Best Known (55−17, 55, s)-Nets in Base 32
(55−17, 55, 4099)-Net over F32 — Constructive and digital
Digital (38, 55, 4099)-net over F32, using
- net defined by OOA [i] based on linear OOA(3255, 4099, F32, 17, 17) (dual of [(4099, 17), 69628, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3255, 32793, F32, 17) (dual of [32793, 32738, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3255, 32795, F32, 17) (dual of [32795, 32740, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- linear OA(3249, 32768, F32, 17) (dual of [32768, 32719, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(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(16) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(3255, 32795, F32, 17) (dual of [32795, 32740, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3255, 32793, F32, 17) (dual of [32793, 32738, 18]-code), using
(55−17, 55, 8192)-Net in Base 32 — Constructive
(38, 55, 8192)-net in base 32, using
- 321 times duplication [i] based on (37, 54, 8192)-net in base 32, using
- base change [i] based on (28, 45, 8192)-net in base 64, using
- 641 times duplication [i] based on (27, 44, 8192)-net in base 64, using
- base change [i] based on digital (16, 33, 8192)-net over F256, using
- net defined by OOA [i] based on linear OOA(25633, 8192, F256, 17, 17) (dual of [(8192, 17), 139231, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using
- net defined by OOA [i] based on linear OOA(25633, 8192, F256, 17, 17) (dual of [(8192, 17), 139231, 18]-NRT-code), using
- base change [i] based on digital (16, 33, 8192)-net over F256, using
- 641 times duplication [i] based on (27, 44, 8192)-net in base 64, using
- base change [i] based on (28, 45, 8192)-net in base 64, using
(55−17, 55, 32795)-Net over F32 — Digital
Digital (38, 55, 32795)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3255, 32795, F32, 17) (dual of [32795, 32740, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- linear OA(3249, 32768, F32, 17) (dual of [32768, 32719, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(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(16) ⊂ Ce(9) [i] based on
(55−17, 55, large)-Net in Base 32 — Upper bound on s
There is no (38, 55, large)-net in base 32, because
- 15 times m-reduction [i] would yield (38, 40, large)-net in base 32, but