Best Known (64−20, 64, s)-Nets in Base 32
(64−20, 64, 3279)-Net over F32 — Constructive and digital
Digital (44, 64, 3279)-net over F32, using
- 321 times duplication [i] based on digital (43, 63, 3279)-net over F32, using
- net defined by OOA [i] based on linear OOA(3263, 3279, F32, 20, 20) (dual of [(3279, 20), 65517, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3263, 32790, F32, 20) (dual of [32790, 32727, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3263, 32791, F32, 20) (dual of [32791, 32728, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(3258, 32768, F32, 20) (dual of [32768, 32710, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3240, 32768, F32, 14) (dual of [32768, 32728, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(19) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3263, 32791, F32, 20) (dual of [32791, 32728, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3263, 32790, F32, 20) (dual of [32790, 32727, 21]-code), using
- net defined by OOA [i] based on linear OOA(3263, 3279, F32, 20, 20) (dual of [(3279, 20), 65517, 21]-NRT-code), using
(64−20, 64, 6554)-Net in Base 32 — Constructive
(44, 64, 6554)-net in base 32, using
- base change [i] based on digital (20, 40, 6554)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 6554, F256, 20, 20) (dual of [(6554, 20), 131040, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(25640, 65540, F256, 20) (dual of [65540, 65500, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, 65541, F256, 20) (dual of [65541, 65501, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- 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(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(25640, 65541, F256, 20) (dual of [65541, 65501, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(25640, 65540, F256, 20) (dual of [65540, 65500, 21]-code), using
- net defined by OOA [i] based on linear OOA(25640, 6554, F256, 20, 20) (dual of [(6554, 20), 131040, 21]-NRT-code), using
(64−20, 64, 32795)-Net over F32 — Digital
Digital (44, 64, 32795)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3264, 32795, F32, 20) (dual of [32795, 32731, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(12) [i] based on
- linear OA(3258, 32768, F32, 20) (dual of [32768, 32710, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3237, 32768, F32, 13) (dual of [32768, 32731, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(19) ⊂ Ce(12) [i] based on
(64−20, 64, large)-Net in Base 32 — Upper bound on s
There is no (44, 64, large)-net in base 32, because
- 18 times m-reduction [i] would yield (44, 46, large)-net in base 32, but