Best Known (54, 54+17, s)-Nets in Base 32
(54, 54+17, 131076)-Net over F32 — Constructive and digital
Digital (54, 71, 131076)-net over F32, using
- net defined by OOA [i] based on linear OOA(3271, 131076, F32, 17, 17) (dual of [(131076, 17), 2228221, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3271, 1048609, F32, 17) (dual of [1048609, 1048538, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- linear OA(3265, 1048576, F32, 17) (dual of [1048576, 1048511, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(3237, 1048576, F32, 10) (dual of [1048576, 1048539, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(326, 33, F32, 6) (dual of [33, 27, 7]-code or 33-arc in PG(5,32)), using
- extended Reed–Solomon code RSe(27,32) [i]
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(3271, 1048609, F32, 17) (dual of [1048609, 1048538, 18]-code), using
(54, 54+17, 262145)-Net in Base 32 — Constructive
(54, 71, 262145)-net in base 32, using
- net defined by OOA [i] based on OOA(3271, 262145, S32, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(3271, 2097161, S32, 17), using
- 1 times code embedding in larger space [i] based on OA(3270, 2097160, S32, 17), using
- discarding parts of the base [i] based on linear OA(12850, 2097160, F128, 17) (dual of [2097160, 2097110, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- discarding parts of the base [i] based on linear OA(12850, 2097160, F128, 17) (dual of [2097160, 2097110, 18]-code), using
- 1 times code embedding in larger space [i] based on OA(3270, 2097160, S32, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(3271, 2097161, S32, 17), using
(54, 54+17, 1048609)-Net over F32 — Digital
Digital (54, 71, 1048609)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3271, 1048609, F32, 17) (dual of [1048609, 1048538, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- linear OA(3265, 1048576, F32, 17) (dual of [1048576, 1048511, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(3237, 1048576, F32, 10) (dual of [1048576, 1048539, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(326, 33, F32, 6) (dual of [33, 27, 7]-code or 33-arc in PG(5,32)), using
- extended Reed–Solomon code RSe(27,32) [i]
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
(54, 54+17, large)-Net in Base 32 — Upper bound on s
There is no (54, 71, large)-net in base 32, because
- 15 times m-reduction [i] would yield (54, 56, large)-net in base 32, but