Best Known (19, 19+7, s)-Nets in Base 32
(19, 19+7, 349528)-Net over F32 — Constructive and digital
Digital (19, 26, 349528)-net over F32, using
- net defined by OOA [i] based on linear OOA(3226, 349528, F32, 7, 7) (dual of [(349528, 7), 2446670, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(3226, 1048585, F32, 7) (dual of [1048585, 1048559, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(3226, 1048586, F32, 7) (dual of [1048586, 1048560, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(3225, 1048577, F32, 7) (dual of [1048577, 1048552, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(3217, 1048577, F32, 5) (dual of [1048577, 1048560, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (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([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3226, 1048586, F32, 7) (dual of [1048586, 1048560, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(3226, 1048585, F32, 7) (dual of [1048585, 1048559, 8]-code), using
(19, 19+7, 1048587)-Net over F32 — Digital
Digital (19, 26, 1048587)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3226, 1048587, F32, 7) (dual of [1048587, 1048561, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(3225, 1048577, F32, 7) (dual of [1048577, 1048552, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(3217, 1048577, F32, 5) (dual of [1048577, 1048560, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(329, 10, F32, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,32)), using
- dual of repetition code with length 10 [i]
- linear OA(321, 10, F32, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- Reed–Solomon code RS(31,32) [i]
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
(19, 19+7, large)-Net in Base 32 — Upper bound on s
There is no (19, 26, large)-net in base 32, because
- 5 times m-reduction [i] would yield (19, 21, large)-net in base 32, but