Best Known (89, 110, s)-Nets in Base 16
(89, 110, 104903)-Net over F16 — Constructive and digital
Digital (89, 110, 104903)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- digital (75, 96, 104858)-net over F16, using
- net defined by OOA [i] based on linear OOA(1696, 104858, F16, 21, 21) (dual of [(104858, 21), 2201922, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(1696, 1048581, F16, 21) (dual of [1048581, 1048485, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(1696, 1048576, F16, 21) (dual of [1048576, 1048480, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1691, 1048576, F16, 20) (dual of [1048576, 1048485, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(20) ⊂ Ce(19) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(1696, 1048581, F16, 21) (dual of [1048581, 1048485, 22]-code), using
- net defined by OOA [i] based on linear OOA(1696, 104858, F16, 21, 21) (dual of [(104858, 21), 2201922, 22]-NRT-code), using
- digital (4, 14, 45)-net over F16, using
(89, 110, 209716)-Net in Base 16 — Constructive
(89, 110, 209716)-net in base 16, using
- net defined by OOA [i] based on OOA(16110, 209716, S16, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(16110, 2097161, S16, 21), using
- 1 times code embedding in larger space [i] based on OA(16109, 2097160, S16, 21), using
- discarding parts of the base [i] based on linear OA(12862, 2097160, F128, 21) (dual of [2097160, 2097098, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(12861, 2097153, F128, 21) (dual of [2097153, 2097092, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(12855, 2097153, F128, 19) (dual of [2097153, 2097098, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding parts of the base [i] based on linear OA(12862, 2097160, F128, 21) (dual of [2097160, 2097098, 22]-code), using
- 1 times code embedding in larger space [i] based on OA(16109, 2097160, S16, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(16110, 2097161, S16, 21), using
(89, 110, 2322078)-Net over F16 — Digital
Digital (89, 110, 2322078)-net over F16, using
(89, 110, large)-Net in Base 16 — Upper bound on s
There is no (89, 110, large)-net in base 16, because
- 19 times m-reduction [i] would yield (89, 91, large)-net in base 16, but