Best Known (45, 45+17, s)-Nets in Base 16
(45, 45+17, 8192)-Net over F16 — Constructive and digital
Digital (45, 62, 8192)-net over F16, using
- net defined by OOA [i] based on linear OOA(1662, 8192, F16, 17, 17) (dual of [(8192, 17), 139202, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1662, 65537, F16, 17) (dual of [65537, 65475, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1662, 65541, F16, 17) (dual of [65541, 65479, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(1661, 65536, F16, 17) (dual of [65536, 65475, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1657, 65536, F16, 15) (dual of [65536, 65479, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(161, 5, F16, 1) (dual of [5, 4, 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 Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(1662, 65541, F16, 17) (dual of [65541, 65479, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1662, 65537, F16, 17) (dual of [65537, 65475, 18]-code), using
(45, 45+17, 33755)-Net over F16 — Digital
Digital (45, 62, 33755)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1662, 33755, F16, 17) (dual of [33755, 33693, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1662, 65541, F16, 17) (dual of [65541, 65479, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(1661, 65536, F16, 17) (dual of [65536, 65475, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1657, 65536, F16, 15) (dual of [65536, 65479, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(161, 5, F16, 1) (dual of [5, 4, 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 Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(1662, 65541, F16, 17) (dual of [65541, 65479, 18]-code), using
(45, 45+17, large)-Net in Base 16 — Upper bound on s
There is no (45, 62, large)-net in base 16, because
- 15 times m-reduction [i] would yield (45, 47, large)-net in base 16, but