Best Known (36−8, 36, s)-Nets in Base 16
(36−8, 36, 262145)-Net over F16 — Constructive and digital
Digital (28, 36, 262145)-net over F16, using
- net defined by OOA [i] based on linear OOA(1636, 262145, F16, 8, 8) (dual of [(262145, 8), 2097124, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(1636, 1048580, F16, 8) (dual of [1048580, 1048544, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(1636, 1048581, F16, 8) (dual of [1048581, 1048545, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(1636, 1048576, F16, 8) (dual of [1048576, 1048540, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1631, 1048576, F16, 7) (dual of [1048576, 1048545, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(1636, 1048581, F16, 8) (dual of [1048581, 1048545, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(1636, 1048580, F16, 8) (dual of [1048580, 1048544, 9]-code), using
(36−8, 36, 1048581)-Net over F16 — Digital
Digital (28, 36, 1048581)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1636, 1048581, F16, 8) (dual of [1048581, 1048545, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(1636, 1048576, F16, 8) (dual of [1048576, 1048540, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1631, 1048576, F16, 7) (dual of [1048576, 1048545, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 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(7) ⊂ Ce(6) [i] based on
(36−8, 36, large)-Net in Base 16 — Upper bound on s
There is no (28, 36, large)-net in base 16, because
- 6 times m-reduction [i] would yield (28, 30, large)-net in base 16, but