Best Known (57, 57+15, s)-Nets in Base 16
(57, 57+15, 149798)-Net over F16 — Constructive and digital
Digital (57, 72, 149798)-net over F16, using
- net defined by OOA [i] based on linear OOA(1672, 149798, F16, 15, 15) (dual of [(149798, 15), 2246898, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1672, 1048587, F16, 15) (dual of [1048587, 1048515, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(1672, 1048588, F16, 15) (dual of [1048588, 1048516, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(1671, 1048577, F16, 15) (dual of [1048577, 1048506, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(1661, 1048577, F16, 13) (dual of [1048577, 1048516, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [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 C([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(1672, 1048588, F16, 15) (dual of [1048588, 1048516, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1672, 1048587, F16, 15) (dual of [1048587, 1048515, 16]-code), using
(57, 57+15, 1048588)-Net over F16 — Digital
Digital (57, 72, 1048588)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1672, 1048588, F16, 15) (dual of [1048588, 1048516, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(1671, 1048577, F16, 15) (dual of [1048577, 1048506, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(1661, 1048577, F16, 13) (dual of [1048577, 1048516, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [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 C([0,7]) ⊂ C([0,6]) [i] based on
(57, 57+15, large)-Net in Base 16 — Upper bound on s
There is no (57, 72, large)-net in base 16, because
- 13 times m-reduction [i] would yield (57, 59, large)-net in base 16, but