Best Known (99−25, 99, s)-Nets in Base 32
(99−25, 99, 87382)-Net over F32 — Constructive and digital
Digital (74, 99, 87382)-net over F32, using
- 321 times duplication [i] based on digital (73, 98, 87382)-net over F32, using
- net defined by OOA [i] based on linear OOA(3298, 87382, F32, 25, 25) (dual of [(87382, 25), 2184452, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3298, 1048585, F32, 25) (dual of [1048585, 1048487, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3298, 1048586, F32, 25) (dual of [1048586, 1048488, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(3297, 1048577, F32, 25) (dual of [1048577, 1048480, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3289, 1048577, F32, 23) (dual of [1048577, 1048488, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3298, 1048586, F32, 25) (dual of [1048586, 1048488, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3298, 1048585, F32, 25) (dual of [1048585, 1048487, 26]-code), using
- net defined by OOA [i] based on linear OOA(3298, 87382, F32, 25, 25) (dual of [(87382, 25), 2184452, 26]-NRT-code), using
(99−25, 99, 787655)-Net over F32 — Digital
Digital (74, 99, 787655)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3299, 787655, F32, 25) (dual of [787655, 787556, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3299, 1048590, F32, 25) (dual of [1048590, 1048491, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3297, 1048576, F32, 25) (dual of [1048576, 1048479, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3285, 1048576, F32, 22) (dual of [1048576, 1048491, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3299, 1048590, F32, 25) (dual of [1048590, 1048491, 26]-code), using
(99−25, 99, large)-Net in Base 32 — Upper bound on s
There is no (74, 99, large)-net in base 32, because
- 23 times m-reduction [i] would yield (74, 76, large)-net in base 32, but