Best Known (50, 69, s)-Nets in Base 16
(50, 69, 7282)-Net over F16 — Constructive and digital
Digital (50, 69, 7282)-net over F16, using
- net defined by OOA [i] based on linear OOA(1669, 7282, F16, 19, 19) (dual of [(7282, 19), 138289, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1669, 65539, F16, 19) (dual of [65539, 65470, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1669, 65540, F16, 19) (dual of [65540, 65471, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(1669, 65536, F16, 19) (dual of [65536, 65467, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1665, 65536, F16, 18) (dual of [65536, 65471, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(160, 4, F16, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(1669, 65540, F16, 19) (dual of [65540, 65471, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1669, 65539, F16, 19) (dual of [65539, 65470, 20]-code), using
(50, 69, 32770)-Net over F16 — Digital
Digital (50, 69, 32770)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(1669, 32770, F16, 2, 19) (dual of [(32770, 2), 65471, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(1669, 65540, F16, 19) (dual of [65540, 65471, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(1669, 65536, F16, 19) (dual of [65536, 65467, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1665, 65536, F16, 18) (dual of [65536, 65471, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(160, 4, F16, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(1669, 65540, F16, 19) (dual of [65540, 65471, 20]-code), using
(50, 69, large)-Net in Base 16 — Upper bound on s
There is no (50, 69, large)-net in base 16, because
- 17 times m-reduction [i] would yield (50, 52, large)-net in base 16, but