Best Known (76−15, 76, s)-Nets in Base 16
(76−15, 76, 149800)-Net over F16 — Constructive and digital
Digital (61, 76, 149800)-net over F16, using
- 161 times duplication [i] based on digital (60, 75, 149800)-net over F16, using
- net defined by OOA [i] based on linear OOA(1675, 149800, F16, 15, 15) (dual of [(149800, 15), 2246925, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1675, 1048601, F16, 15) (dual of [1048601, 1048526, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [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(1651, 1048577, F16, 11) (dual of [1048577, 1048526, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,7]) ⊂ C([0,5]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(1675, 1048601, F16, 15) (dual of [1048601, 1048526, 16]-code), using
- net defined by OOA [i] based on linear OOA(1675, 149800, F16, 15, 15) (dual of [(149800, 15), 2246925, 16]-NRT-code), using
(76−15, 76, 299593)-Net in Base 16 — Constructive
(61, 76, 299593)-net in base 16, using
- net defined by OOA [i] based on OOA(1676, 299593, S16, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(1676, 2097152, S16, 15), using
- discarding factors based on OA(1676, 2097155, S16, 15), using
- discarding parts of the base [i] based on linear OA(12843, 2097155, F128, 15) (dual of [2097155, 2097112, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(14) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(12843, 2097155, F128, 15) (dual of [2097155, 2097112, 16]-code), using
- discarding factors based on OA(1676, 2097155, S16, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(1676, 2097152, S16, 15), using
(76−15, 76, 1386820)-Net over F16 — Digital
Digital (61, 76, 1386820)-net over F16, using
(76−15, 76, large)-Net in Base 16 — Upper bound on s
There is no (61, 76, large)-net in base 16, because
- 13 times m-reduction [i] would yield (61, 63, large)-net in base 16, but