Best Known (78, 78+21, s)-Nets in Base 16
(78, 78+21, 104859)-Net over F16 — Constructive and digital
Digital (78, 99, 104859)-net over F16, using
- 161 times duplication [i] based on digital (77, 98, 104859)-net over F16, using
- net defined by OOA [i] based on linear OOA(1698, 104859, F16, 21, 21) (dual of [(104859, 21), 2201941, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(1698, 1048591, F16, 21) (dual of [1048591, 1048493, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(1698, 1048593, F16, 21) (dual of [1048593, 1048495, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(1696, 1048576, F16, 21) (dual of [1048576, 1048480, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1681, 1048576, F16, 18) (dual of [1048576, 1048495, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(1698, 1048593, F16, 21) (dual of [1048593, 1048495, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(1698, 1048591, F16, 21) (dual of [1048591, 1048493, 22]-code), using
- net defined by OOA [i] based on linear OOA(1698, 104859, F16, 21, 21) (dual of [(104859, 21), 2201941, 22]-NRT-code), using
(78, 78+21, 858704)-Net over F16 — Digital
Digital (78, 99, 858704)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1699, 858704, F16, 21) (dual of [858704, 858605, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(1699, 1048594, F16, 21) (dual of [1048594, 1048495, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1698, 1048593, F16, 21) (dual of [1048593, 1048495, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(1696, 1048576, F16, 21) (dual of [1048576, 1048480, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1681, 1048576, F16, 18) (dual of [1048576, 1048495, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(20) ⊂ Ce(17) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(1698, 1048593, F16, 21) (dual of [1048593, 1048495, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(1699, 1048594, F16, 21) (dual of [1048594, 1048495, 22]-code), using
(78, 78+21, large)-Net in Base 16 — Upper bound on s
There is no (78, 99, large)-net in base 16, because
- 19 times m-reduction [i] would yield (78, 80, large)-net in base 16, but