Best Known (42−9, 42, s)-Nets in Base 16
(42−9, 42, 262147)-Net over F16 — Constructive and digital
Digital (33, 42, 262147)-net over F16, using
- net defined by OOA [i] based on linear OOA(1642, 262147, F16, 9, 9) (dual of [(262147, 9), 2359281, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(1642, 1048589, F16, 9) (dual of [1048589, 1048547, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(1641, 1048577, F16, 9) (dual of [1048577, 1048536, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(1631, 1048577, F16, 7) (dual of [1048577, 1048546, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(1611, 12, F16, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,16)), using
- dual of repetition code with length 12 [i]
- linear OA(161, 12, F16, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, 16, F16, 1) (dual of [16, 15, 2]-code), using
- Reed–Solomon code RS(15,16) [i]
- discarding factors / shortening the dual code based on linear OA(161, 16, F16, 1) (dual of [16, 15, 2]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(1642, 1048589, F16, 9) (dual of [1048589, 1048547, 10]-code), using
(42−9, 42, 1048589)-Net over F16 — Digital
Digital (33, 42, 1048589)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1642, 1048589, F16, 9) (dual of [1048589, 1048547, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(1641, 1048577, F16, 9) (dual of [1048577, 1048536, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(1631, 1048577, F16, 7) (dual of [1048577, 1048546, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(1611, 12, F16, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,16)), using
- dual of repetition code with length 12 [i]
- linear OA(161, 12, F16, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, 16, F16, 1) (dual of [16, 15, 2]-code), using
- Reed–Solomon code RS(15,16) [i]
- discarding factors / shortening the dual code based on linear OA(161, 16, F16, 1) (dual of [16, 15, 2]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
(42−9, 42, large)-Net in Base 16 — Upper bound on s
There is no (33, 42, large)-net in base 16, because
- 7 times m-reduction [i] would yield (33, 35, large)-net in base 16, but