Best Known (56, 75, s)-Nets in Base 32
(56, 75, 116509)-Net over F32 — Constructive and digital
Digital (56, 75, 116509)-net over F32, using
- 321 times duplication [i] based on digital (55, 74, 116509)-net over F32, using
- net defined by OOA [i] based on linear OOA(3274, 116509, F32, 19, 19) (dual of [(116509, 19), 2213597, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3274, 1048582, F32, 19) (dual of [1048582, 1048508, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3274, 1048586, F32, 19) (dual of [1048586, 1048512, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(3273, 1048577, F32, 19) (dual of [1048577, 1048504, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(3265, 1048577, F32, 17) (dual of [1048577, 1048512, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(321, 9, F32, 1) (dual of [9, 8, 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([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3274, 1048586, F32, 19) (dual of [1048586, 1048512, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3274, 1048582, F32, 19) (dual of [1048582, 1048508, 20]-code), using
- net defined by OOA [i] based on linear OOA(3274, 116509, F32, 19, 19) (dual of [(116509, 19), 2213597, 20]-NRT-code), using
(56, 75, 824914)-Net over F32 — Digital
Digital (56, 75, 824914)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3275, 824914, F32, 19) (dual of [824914, 824839, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3275, 1048590, F32, 19) (dual of [1048590, 1048515, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3273, 1048576, F32, 19) (dual of [1048576, 1048503, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3261, 1048576, F32, 16) (dual of [1048576, 1048515, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3275, 1048590, F32, 19) (dual of [1048590, 1048515, 20]-code), using
(56, 75, large)-Net in Base 32 — Upper bound on s
There is no (56, 75, large)-net in base 32, because
- 17 times m-reduction [i] would yield (56, 58, large)-net in base 32, but