Best Known (38−8, 38, s)-Nets in Base 16
(38−8, 38, 262148)-Net over F16 — Constructive and digital
Digital (30, 38, 262148)-net over F16, using
- net defined by OOA [i] based on linear OOA(1638, 262148, F16, 8, 8) (dual of [(262148, 8), 2097146, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(1638, 1048592, F16, 8) (dual of [1048592, 1048554, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(1638, 1048593, F16, 8) (dual of [1048593, 1048555, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(1636, 1048576, F16, 8) (dual of [1048576, 1048540, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1621, 1048576, F16, 5) (dual of [1048576, 1048555, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(162, 17, F16, 2) (dual of [17, 15, 3]-code or 17-arc in PG(1,16)), using
- extended Reed–Solomon code RSe(15,16) [i]
- Hamming code H(2,16) [i]
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(1638, 1048593, F16, 8) (dual of [1048593, 1048555, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(1638, 1048592, F16, 8) (dual of [1048592, 1048554, 9]-code), using
(38−8, 38, 1048593)-Net over F16 — Digital
Digital (30, 38, 1048593)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1638, 1048593, F16, 8) (dual of [1048593, 1048555, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(1636, 1048576, F16, 8) (dual of [1048576, 1048540, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1621, 1048576, F16, 5) (dual of [1048576, 1048555, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(162, 17, F16, 2) (dual of [17, 15, 3]-code or 17-arc in PG(1,16)), using
- extended Reed–Solomon code RSe(15,16) [i]
- Hamming code H(2,16) [i]
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
(38−8, 38, large)-Net in Base 16 — Upper bound on s
There is no (30, 38, large)-net in base 16, because
- 6 times m-reduction [i] would yield (30, 32, large)-net in base 16, but