Best Known (13, 13+7, s)-Nets in Base 32
(13, 13+7, 10925)-Net over F32 — Constructive and digital
Digital (13, 20, 10925)-net over F32, using
- net defined by OOA [i] based on linear OOA(3220, 10925, F32, 7, 7) (dual of [(10925, 7), 76455, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(3220, 32776, F32, 7) (dual of [32776, 32756, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(3219, 32769, F32, 7) (dual of [32769, 32750, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(3213, 32769, F32, 5) (dual of [32769, 32756, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (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([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(3220, 32776, F32, 7) (dual of [32776, 32756, 8]-code), using
(13, 13+7, 32777)-Net over F32 — Digital
Digital (13, 20, 32777)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3220, 32777, F32, 7) (dual of [32777, 32757, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(3219, 32769, F32, 7) (dual of [32769, 32750, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(3213, 32769, F32, 5) (dual of [32769, 32756, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(327, 8, F32, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,32)), using
- dual of repetition code with length 8 [i]
- linear OA(321, 8, F32, 1) (dual of [8, 7, 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
(13, 13+7, large)-Net in Base 32 — Upper bound on s
There is no (13, 20, large)-net in base 32, because
- 5 times m-reduction [i] would yield (13, 15, large)-net in base 32, but