Best Known (79−20, 79, s)-Nets in Base 16
(79−20, 79, 13107)-Net over F16 — Constructive and digital
Digital (59, 79, 13107)-net over F16, using
- 161 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(1678, 13107, F16, 20, 20) (dual of [(13107, 20), 262062, 21]-NRT-code), using
(79−20, 79, 83147)-Net over F16 — Digital
Digital (59, 79, 83147)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1679, 83147, F16, 20) (dual of [83147, 83068, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1679, 131077, F16, 20) (dual of [131077, 130998, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1678, 131076, F16, 20) (dual of [131076, 130998, 21]-code), using
- trace code [i] based on linear OA(25639, 65538, F256, 20) (dual of [65538, 65499, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [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]
- linear OA(25637, 65536, F256, 19) (dual of [65536, 65499, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- trace code [i] based on linear OA(25639, 65538, F256, 20) (dual of [65538, 65499, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1678, 131076, F16, 20) (dual of [131076, 130998, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1679, 131077, F16, 20) (dual of [131077, 130998, 21]-code), using
(79−20, 79, large)-Net in Base 16 — Upper bound on s
There is no (59, 79, large)-net in base 16, because
- 18 times m-reduction [i] would yield (59, 61, large)-net in base 16, but