Best Known (52, 52+13, s)-Nets in Base 16
(52, 52+13, 174766)-Net over F16 — Constructive and digital
Digital (52, 65, 174766)-net over F16, using
- net defined by OOA [i] based on linear OOA(1665, 174766, F16, 13, 13) (dual of [(174766, 13), 2271893, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1665, 1048597, F16, 13) (dual of [1048597, 1048532, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1665, 1048601, F16, 13) (dual of [1048601, 1048536, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- 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(1641, 1048577, F16, 9) (dual of [1048577, 1048536, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(164, 24, F16, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,16)), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(1665, 1048601, F16, 13) (dual of [1048601, 1048536, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1665, 1048597, F16, 13) (dual of [1048597, 1048532, 14]-code), using
(52, 52+13, 349525)-Net in Base 16 — Constructive
(52, 65, 349525)-net in base 16, using
- net defined by OOA [i] based on OOA(1665, 349525, S16, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1665, 2097151, S16, 13), using
- discarding factors based on OA(1665, 2097155, S16, 13), using
- discarding parts of the base [i] based on linear OA(12837, 2097155, F128, 13) (dual of [2097155, 2097118, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(12837, 2097155, F128, 13) (dual of [2097155, 2097118, 14]-code), using
- discarding factors based on OA(1665, 2097155, S16, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1665, 2097151, S16, 13), using
(52, 52+13, 1173786)-Net over F16 — Digital
Digital (52, 65, 1173786)-net over F16, using
(52, 52+13, large)-Net in Base 16 — Upper bound on s
There is no (52, 65, large)-net in base 16, because
- 11 times m-reduction [i] would yield (52, 54, large)-net in base 16, but