Best Known (48, 48+13, s)-Nets in Base 16
(48, 48+13, 174763)-Net over F16 — Constructive and digital
Digital (48, 61, 174763)-net over F16, using
- net defined by OOA [i] based on linear OOA(1661, 174763, F16, 13, 13) (dual of [(174763, 13), 2271858, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1661, 1048579, F16, 13) (dual of [1048579, 1048518, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1661, 1048581, F16, 13) (dual of [1048581, 1048520, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(1661, 1048576, F16, 13) (dual of [1048576, 1048515, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(1656, 1048576, F16, 12) (dual of [1048576, 1048520, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(1661, 1048581, F16, 13) (dual of [1048581, 1048520, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1661, 1048579, F16, 13) (dual of [1048579, 1048518, 14]-code), using
(48, 48+13, 1048581)-Net over F16 — Digital
Digital (48, 61, 1048581)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1661, 1048581, F16, 13) (dual of [1048581, 1048520, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(1661, 1048576, F16, 13) (dual of [1048576, 1048515, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(1656, 1048576, F16, 12) (dual of [1048576, 1048520, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
(48, 48+13, large)-Net in Base 16 — Upper bound on s
There is no (48, 61, large)-net in base 16, because
- 11 times m-reduction [i] would yield (48, 50, large)-net in base 16, but