Best Known (72−18, 72, s)-Nets in Base 32
(72−18, 72, 116510)-Net over F32 — Constructive and digital
Digital (54, 72, 116510)-net over F32, using
- 321 times duplication [i] based on digital (53, 71, 116510)-net over F32, using
- net defined by OOA [i] based on linear OOA(3271, 116510, F32, 18, 18) (dual of [(116510, 18), 2097109, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3271, 1048590, F32, 18) (dual of [1048590, 1048519, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(3269, 1048576, F32, 18) (dual of [1048576, 1048507, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(3257, 1048576, F32, 15) (dual of [1048576, 1048519, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(322, 14, F32, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- OA 9-folding and stacking [i] based on linear OA(3271, 1048590, F32, 18) (dual of [1048590, 1048519, 19]-code), using
- net defined by OOA [i] based on linear OOA(3271, 116510, F32, 18, 18) (dual of [(116510, 18), 2097109, 19]-NRT-code), using
(72−18, 72, 1047797)-Net over F32 — Digital
Digital (54, 72, 1047797)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3272, 1047797, F32, 18) (dual of [1047797, 1047725, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3272, 1048595, F32, 18) (dual of [1048595, 1048523, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(3269, 1048576, F32, 18) (dual of [1048576, 1048507, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(3253, 1048576, F32, 14) (dual of [1048576, 1048523, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(323, 19, F32, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,32) or 19-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 Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3272, 1048595, F32, 18) (dual of [1048595, 1048523, 19]-code), using
(72−18, 72, large)-Net in Base 32 — Upper bound on s
There is no (54, 72, large)-net in base 32, because
- 16 times m-reduction [i] would yield (54, 56, large)-net in base 32, but