Best Known (64, 64+31, s)-Nets in Base 32
(64, 64+31, 2185)-Net over F32 — Constructive and digital
Digital (64, 95, 2185)-net over F32, using
- 323 times duplication [i] based on digital (61, 92, 2185)-net over F32, using
- net defined by OOA [i] based on linear OOA(3292, 2185, F32, 31, 31) (dual of [(2185, 31), 67643, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3292, 32776, F32, 31) (dual of [32776, 32684, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,14]) [i] based on
- linear OA(3291, 32769, F32, 31) (dual of [32769, 32678, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3285, 32769, F32, 29) (dual of [32769, 32684, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(321, 7, F32, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,15]) ⊂ C([0,14]) [i] based on
- OOA 15-folding and stacking with additional row [i] based on linear OA(3292, 32776, F32, 31) (dual of [32776, 32684, 32]-code), using
- net defined by OOA [i] based on linear OOA(3292, 2185, F32, 31, 31) (dual of [(2185, 31), 67643, 32]-NRT-code), using
(64, 64+31, 28464)-Net over F32 — Digital
Digital (64, 95, 28464)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3295, 28464, F32, 31) (dual of [28464, 28369, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3295, 32787, F32, 31) (dual of [32787, 32692, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(3291, 32768, F32, 31) (dual of [32768, 32677, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3276, 32768, F32, 26) (dual of [32768, 32692, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(30) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3295, 32787, F32, 31) (dual of [32787, 32692, 32]-code), using
(64, 64+31, large)-Net in Base 32 — Upper bound on s
There is no (64, 95, large)-net in base 32, because
- 29 times m-reduction [i] would yield (64, 66, large)-net in base 32, but