Best Known (67−14, 67, s)-Nets in Base 16
(67−14, 67, 149798)-Net over F16 — Constructive and digital
Digital (53, 67, 149798)-net over F16, using
- net defined by OOA [i] based on linear OOA(1667, 149798, F16, 14, 14) (dual of [(149798, 14), 2097105, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(1667, 1048586, F16, 14) (dual of [1048586, 1048519, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1667, 1048587, F16, 14) (dual of [1048587, 1048520, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(1666, 1048576, F16, 14) (dual of [1048576, 1048510, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(161, 11, F16, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(1667, 1048587, F16, 14) (dual of [1048587, 1048520, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(1667, 1048586, F16, 14) (dual of [1048586, 1048519, 15]-code), using
(67−14, 67, 1048587)-Net over F16 — Digital
Digital (53, 67, 1048587)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1667, 1048587, F16, 14) (dual of [1048587, 1048520, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(1666, 1048576, F16, 14) (dual of [1048576, 1048510, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(161, 11, F16, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
(67−14, 67, large)-Net in Base 16 — Upper bound on s
There is no (53, 67, large)-net in base 16, because
- 12 times m-reduction [i] would yield (53, 55, large)-net in base 16, but