Best Known (22, 22+12, s)-Nets in Base 16
(22, 22+12, 683)-Net over F16 — Constructive and digital
Digital (22, 34, 683)-net over F16, using
- net defined by OOA [i] based on linear OOA(1634, 683, F16, 12, 12) (dual of [(683, 12), 8162, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(1634, 4098, F16, 12) (dual of [4098, 4064, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(1634, 4099, F16, 12) (dual of [4099, 4065, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(1634, 4096, F16, 12) (dual of [4096, 4062, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1631, 4096, F16, 11) (dual of [4096, 4065, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(160, 3, F16, 0) (dual of [3, 3, 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(1634, 4099, F16, 12) (dual of [4099, 4065, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(1634, 4098, F16, 12) (dual of [4098, 4064, 13]-code), using
(22, 22+12, 2836)-Net over F16 — Digital
Digital (22, 34, 2836)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1634, 2836, F16, 12) (dual of [2836, 2802, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(1634, 4096, F16, 12) (dual of [4096, 4062, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(1634, 4096, F16, 12) (dual of [4096, 4062, 13]-code), using
(22, 22+12, 1328851)-Net in Base 16 — Upper bound on s
There is no (22, 34, 1328852)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 87112 490632 303066 371399 937970 926489 620556 > 1634 [i]