Best Known (56, 71, s)-Nets in Base 16
(56, 71, 149797)-Net over F16 — Constructive and digital
Digital (56, 71, 149797)-net over F16, using
- net defined by OOA [i] based on linear OOA(1671, 149797, F16, 15, 15) (dual of [(149797, 15), 2246884, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1671, 1048580, F16, 15) (dual of [1048580, 1048509, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(1671, 1048581, F16, 15) (dual of [1048581, 1048510, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(1671, 1048576, F16, 15) (dual of [1048576, 1048505, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(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(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(1671, 1048581, F16, 15) (dual of [1048581, 1048510, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1671, 1048580, F16, 15) (dual of [1048580, 1048509, 16]-code), using
(56, 71, 1048581)-Net over F16 — Digital
Digital (56, 71, 1048581)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1671, 1048581, F16, 15) (dual of [1048581, 1048510, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(1671, 1048576, F16, 15) (dual of [1048576, 1048505, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(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(14) ⊂ Ce(13) [i] based on
(56, 71, large)-Net in Base 16 — Upper bound on s
There is no (56, 71, large)-net in base 16, because
- 13 times m-reduction [i] would yield (56, 58, large)-net in base 16, but