Best Known (83−21, 83, s)-Nets in Base 32
(83−21, 83, 104858)-Net over F32 — Constructive and digital
Digital (62, 83, 104858)-net over F32, using
- 321 times duplication [i] based on digital (61, 82, 104858)-net over F32, using
- net defined by OOA [i] based on linear OOA(3282, 104858, F32, 21, 21) (dual of [(104858, 21), 2201936, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3282, 1048581, F32, 21) (dual of [1048581, 1048499, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3282, 1048586, F32, 21) (dual of [1048586, 1048504, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3281, 1048577, F32, 21) (dual of [1048577, 1048496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 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(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([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3282, 1048586, F32, 21) (dual of [1048586, 1048504, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3282, 1048581, F32, 21) (dual of [1048581, 1048499, 22]-code), using
- net defined by OOA [i] based on linear OOA(3282, 104858, F32, 21, 21) (dual of [(104858, 21), 2201936, 22]-NRT-code), using
(83−21, 83, 801234)-Net over F32 — Digital
Digital (62, 83, 801234)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3283, 801234, F32, 21) (dual of [801234, 801151, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3283, 1048590, F32, 21) (dual of [1048590, 1048507, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(3281, 1048576, F32, 21) (dual of [1048576, 1048495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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(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(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(3283, 1048590, F32, 21) (dual of [1048590, 1048507, 22]-code), using
(83−21, 83, large)-Net in Base 32 — Upper bound on s
There is no (62, 83, large)-net in base 32, because
- 19 times m-reduction [i] would yield (62, 64, large)-net in base 32, but