Best Known (45−12, 45, s)-Nets in Base 16
(45−12, 45, 10923)-Net over F16 — Constructive and digital
Digital (33, 45, 10923)-net over F16, using
- net defined by OOA [i] based on linear OOA(1645, 10923, F16, 12, 12) (dual of [(10923, 12), 131031, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(1645, 65538, F16, 12) (dual of [65538, 65493, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(1645, 65540, F16, 12) (dual of [65540, 65495, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(1645, 65536, F16, 12) (dual of [65536, 65491, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1641, 65536, F16, 11) (dual of [65536, 65495, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(1645, 65540, F16, 12) (dual of [65540, 65495, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(1645, 65538, F16, 12) (dual of [65538, 65493, 13]-code), using
(45−12, 45, 59976)-Net over F16 — Digital
Digital (33, 45, 59976)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1645, 59976, F16, 12) (dual of [59976, 59931, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(1645, 65536, F16, 12) (dual of [65536, 65491, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(1645, 65536, F16, 12) (dual of [65536, 65491, 13]-code), using
(45−12, 45, large)-Net in Base 16 — Upper bound on s
There is no (33, 45, large)-net in base 16, because
- 10 times m-reduction [i] would yield (33, 35, large)-net in base 16, but