Best Known (44, 55, s)-Nets in Base 16
(44, 55, 209720)-Net over F16 — Constructive and digital
Digital (44, 55, 209720)-net over F16, using
- net defined by OOA [i] based on linear OOA(1655, 209720, F16, 11, 11) (dual of [(209720, 11), 2306865, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(1655, 1048601, F16, 11) (dual of [1048601, 1048546, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- 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(1631, 1048577, F16, 7) (dual of [1048577, 1048546, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,5]) ⊂ C([0,3]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(1655, 1048601, F16, 11) (dual of [1048601, 1048546, 12]-code), using
(44, 55, 419430)-Net in Base 16 — Constructive
(44, 55, 419430)-net in base 16, using
- net defined by OOA [i] based on OOA(1655, 419430, S16, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(1655, 2097151, S16, 11), using
- discarding factors based on OA(1655, 2097155, S16, 11), using
- discarding parts of the base [i] based on linear OA(12831, 2097155, F128, 11) (dual of [2097155, 2097124, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(12831, 2097152, F128, 11) (dual of [2097152, 2097121, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- discarding parts of the base [i] based on linear OA(12831, 2097155, F128, 11) (dual of [2097155, 2097124, 12]-code), using
- discarding factors based on OA(1655, 2097155, S16, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(1655, 2097151, S16, 11), using
(44, 55, 1266329)-Net over F16 — Digital
Digital (44, 55, 1266329)-net over F16, using
(44, 55, large)-Net in Base 16 — Upper bound on s
There is no (44, 55, large)-net in base 16, because
- 9 times m-reduction [i] would yield (44, 46, large)-net in base 16, but