Best Known (37, 37+17, s)-Nets in Base 32
(37, 37+17, 4098)-Net over F32 — Constructive and digital
Digital (37, 54, 4098)-net over F32, using
- 321 times duplication [i] based on digital (36, 53, 4098)-net over F32, using
- net defined by OOA [i] based on linear OOA(3253, 4098, F32, 17, 17) (dual of [(4098, 17), 69613, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3253, 32785, F32, 17) (dual of [32785, 32732, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3253, 32787, F32, 17) (dual of [32787, 32734, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [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(3234, 32768, F32, 12) (dual of [32768, 32734, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(16) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(3253, 32787, F32, 17) (dual of [32787, 32734, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3253, 32785, F32, 17) (dual of [32785, 32732, 18]-code), using
- net defined by OOA [i] based on linear OOA(3253, 4098, F32, 17, 17) (dual of [(4098, 17), 69613, 18]-NRT-code), using
(37, 37+17, 8192)-Net in Base 32 — Constructive
(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
(37, 37+17, 32792)-Net over F32 — Digital
Digital (37, 54, 32792)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3254, 32792, F32, 17) (dual of [32792, 32738, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(3249, 32769, F32, 17) (dual of [32769, 32720, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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(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([0,8]) ⊂ C([0,5]) [i] based on
(37, 37+17, large)-Net in Base 32 — Upper bound on s
There is no (37, 54, large)-net in base 32, because
- 15 times m-reduction [i] would yield (37, 39, large)-net in base 32, but