Best Known (38−13, 38, s)-Nets in Base 32
(38−13, 38, 5462)-Net over F32 — Constructive and digital
Digital (25, 38, 5462)-net over F32, using
- net defined by OOA [i] based on linear OOA(3238, 5462, F32, 13, 13) (dual of [(5462, 13), 70968, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3238, 32773, F32, 13) (dual of [32773, 32735, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3238, 32776, F32, 13) (dual of [32776, 32738, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(3237, 32769, F32, 13) (dual of [32769, 32732, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(3231, 32769, F32, 11) (dual of [32769, 32738, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3238, 32776, F32, 13) (dual of [32776, 32738, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3238, 32773, F32, 13) (dual of [32773, 32735, 14]-code), using
(38−13, 38, 18294)-Net over F32 — Digital
Digital (25, 38, 18294)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3238, 18294, F32, 13) (dual of [18294, 18256, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3238, 32776, F32, 13) (dual of [32776, 32738, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(3237, 32769, F32, 13) (dual of [32769, 32732, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(3231, 32769, F32, 11) (dual of [32769, 32738, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3238, 32776, F32, 13) (dual of [32776, 32738, 14]-code), using
(38−13, 38, large)-Net in Base 32 — Upper bound on s
There is no (25, 38, large)-net in base 32, because
- 11 times m-reduction [i] would yield (25, 27, large)-net in base 32, but