Best Known (62−13, 62, s)-Nets in Base 16
(62−13, 62, 174764)-Net over F16 — Constructive and digital
Digital (49, 62, 174764)-net over F16, using
- net defined by OOA [i] based on linear OOA(1662, 174764, F16, 13, 13) (dual of [(174764, 13), 2271870, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1662, 1048585, F16, 13) (dual of [1048585, 1048523, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1662, 1048588, F16, 13) (dual of [1048588, 1048526, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(1661, 1048577, F16, 13) (dual of [1048577, 1048516, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1651, 1048577, F16, 11) (dual of [1048577, 1048526, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(161, 11, F16, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 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(1662, 1048588, F16, 13) (dual of [1048588, 1048526, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1662, 1048585, F16, 13) (dual of [1048585, 1048523, 14]-code), using
(62−13, 62, 1048588)-Net over F16 — Digital
Digital (49, 62, 1048588)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1662, 1048588, F16, 13) (dual of [1048588, 1048526, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(1661, 1048577, F16, 13) (dual of [1048577, 1048516, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1651, 1048577, F16, 11) (dual of [1048577, 1048526, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(161, 11, F16, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 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
(62−13, 62, large)-Net in Base 16 — Upper bound on s
There is no (49, 62, large)-net in base 16, because
- 11 times m-reduction [i] would yield (49, 51, large)-net in base 16, but