Best Known (18, 18+7, s)-Nets in Base 16
(18, 18+7, 21846)-Net over F16 — Constructive and digital
Digital (18, 25, 21846)-net over F16, using
- net defined by OOA [i] based on linear OOA(1625, 21846, F16, 7, 7) (dual of [(21846, 7), 152897, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1625, 65539, F16, 7) (dual of [65539, 65514, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(1625, 65540, F16, 7) (dual of [65540, 65515, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(1625, 65536, F16, 7) (dual of [65536, 65511, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(1621, 65536, F16, 6) (dual of [65536, 65515, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(1625, 65540, F16, 7) (dual of [65540, 65515, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1625, 65539, F16, 7) (dual of [65539, 65514, 8]-code), using
(18, 18+7, 65540)-Net over F16 — Digital
Digital (18, 25, 65540)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1625, 65540, F16, 7) (dual of [65540, 65515, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(1625, 65536, F16, 7) (dual of [65536, 65511, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(1621, 65536, F16, 6) (dual of [65536, 65515, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
(18, 18+7, large)-Net in Base 16 — Upper bound on s
There is no (18, 25, large)-net in base 16, because
- 5 times m-reduction [i] would yield (18, 20, large)-net in base 16, but