Best Known (78−20, 78, s)-Nets in Base 16
(78−20, 78, 13107)-Net over F16 — Constructive and digital
Digital (58, 78, 13107)-net over F16, using
- net defined by OOA [i] based on linear OOA(1678, 13107, F16, 20, 20) (dual of [(13107, 20), 262062, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(1678, 131070, F16, 20) (dual of [131070, 130992, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1678, 131072, F16, 20) (dual of [131072, 130994, 21]-code), using
- trace code [i] based on linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- trace code [i] based on linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1678, 131072, F16, 20) (dual of [131072, 130994, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(1678, 131070, F16, 20) (dual of [131070, 130992, 21]-code), using
(78−20, 78, 71276)-Net over F16 — Digital
Digital (58, 78, 71276)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1678, 71276, F16, 20) (dual of [71276, 71198, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1678, 131072, F16, 20) (dual of [131072, 130994, 21]-code), using
- trace code [i] based on linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- trace code [i] based on linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1678, 131072, F16, 20) (dual of [131072, 130994, 21]-code), using
(78−20, 78, large)-Net in Base 16 — Upper bound on s
There is no (58, 78, large)-net in base 16, because
- 18 times m-reduction [i] would yield (58, 60, large)-net in base 16, but