Best Known (72, 72+34, s)-Nets in Base 32
(72, 72+34, 1929)-Net over F32 — Constructive and digital
Digital (72, 106, 1929)-net over F32, using
- 1 times m-reduction [i] based on digital (72, 107, 1929)-net over F32, using
- net defined by OOA [i] based on linear OOA(32107, 1929, F32, 35, 35) (dual of [(1929, 35), 67408, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(32107, 32794, F32, 35) (dual of [32794, 32687, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(32107, 32796, F32, 35) (dual of [32796, 32689, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(26) [i] based on
- linear OA(32100, 32768, F32, 35) (dual of [32768, 32668, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3279, 32768, F32, 27) (dual of [32768, 32689, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(327, 28, F32, 7) (dual of [28, 21, 8]-code or 28-arc in PG(6,32)), using
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- Reed–Solomon code RS(25,32) [i]
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- construction X applied to Ce(34) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(32107, 32796, F32, 35) (dual of [32796, 32689, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(32107, 32794, F32, 35) (dual of [32794, 32687, 36]-code), using
- net defined by OOA [i] based on linear OOA(32107, 1929, F32, 35, 35) (dual of [(1929, 35), 67408, 36]-NRT-code), using
(72, 72+34, 32802)-Net over F32 — Digital
Digital (72, 106, 32802)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32106, 32802, F32, 34) (dual of [32802, 32696, 35]-code), using
- construction XX applied to Ce(33) ⊂ Ce(24) ⊂ Ce(23) [i] based on
- linear OA(3297, 32768, F32, 34) (dual of [32768, 32671, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3273, 32768, F32, 25) (dual of [32768, 32695, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3270, 32768, F32, 24) (dual of [32768, 32698, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(328, 33, F32, 8) (dual of [33, 25, 9]-code or 33-arc in PG(7,32)), using
- extended Reed–Solomon code RSe(25,32) [i]
- linear OA(320, 1, F32, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(33) ⊂ Ce(24) ⊂ Ce(23) [i] based on
(72, 72+34, large)-Net in Base 32 — Upper bound on s
There is no (72, 106, large)-net in base 32, because
- 32 times m-reduction [i] would yield (72, 74, large)-net in base 32, but