Best Known (44, 44+21, s)-Nets in Base 32
(44, 44+21, 3278)-Net over F32 — Constructive and digital
Digital (44, 65, 3278)-net over F32, using
- 321 times duplication [i] based on digital (43, 64, 3278)-net over F32, using
- net defined by OOA [i] based on linear OOA(3264, 3278, F32, 21, 21) (dual of [(3278, 21), 68774, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3264, 32781, F32, 21) (dual of [32781, 32717, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3264, 32784, F32, 21) (dual of [32784, 32720, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [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(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(323, 15, F32, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,32) or 15-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3264, 32784, F32, 21) (dual of [32784, 32720, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3264, 32781, F32, 21) (dual of [32781, 32717, 22]-code), using
- net defined by OOA [i] based on linear OOA(3264, 3278, F32, 21, 21) (dual of [(3278, 21), 68774, 22]-NRT-code), using
(44, 44+21, 30040)-Net over F32 — Digital
Digital (44, 65, 30040)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3265, 30040, F32, 21) (dual of [30040, 29975, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3265, 32787, F32, 21) (dual of [32787, 32722, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(3261, 32768, F32, 21) (dual of [32768, 32707, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(3246, 32768, F32, 16) (dual of [32768, 32722, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(20) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3265, 32787, F32, 21) (dual of [32787, 32722, 22]-code), using
(44, 44+21, large)-Net in Base 32 — Upper bound on s
There is no (44, 65, large)-net in base 32, because
- 19 times m-reduction [i] would yield (44, 46, large)-net in base 32, but