Best Known (68−13, 68, s)-Nets in Base 16
(68−13, 68, 174787)-Net over F16 — Constructive and digital
Digital (55, 68, 174787)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- digital (48, 61, 174763)-net over F16, using
- net defined by OOA [i] based on linear OOA(1661, 174763, F16, 13, 13) (dual of [(174763, 13), 2271858, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1661, 1048579, F16, 13) (dual of [1048579, 1048518, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1661, 1048581, F16, 13) (dual of [1048581, 1048520, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(1661, 1048576, F16, 13) (dual of [1048576, 1048515, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(1656, 1048576, F16, 12) (dual of [1048576, 1048520, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(1661, 1048581, F16, 13) (dual of [1048581, 1048520, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1661, 1048579, F16, 13) (dual of [1048579, 1048518, 14]-code), using
- net defined by OOA [i] based on linear OOA(1661, 174763, F16, 13, 13) (dual of [(174763, 13), 2271858, 14]-NRT-code), using
- digital (1, 7, 24)-net over F16, using
(68−13, 68, 349526)-Net in Base 16 — Constructive
(55, 68, 349526)-net in base 16, using
- 161 times duplication [i] based on (54, 67, 349526)-net in base 16, using
- net defined by OOA [i] based on OOA(1667, 349526, S16, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1667, 2097157, S16, 13), using
- discarding factors based on OA(1667, 2097160, S16, 13), using
- discarding parts of the base [i] based on linear OA(12838, 2097160, F128, 13) (dual of [2097160, 2097122, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding parts of the base [i] based on linear OA(12838, 2097160, F128, 13) (dual of [2097160, 2097122, 14]-code), using
- discarding factors based on OA(1667, 2097160, S16, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1667, 2097157, S16, 13), using
- net defined by OOA [i] based on OOA(1667, 349526, S16, 13, 13), using
(68−13, 68, 2347566)-Net over F16 — Digital
Digital (55, 68, 2347566)-net over F16, using
(68−13, 68, large)-Net in Base 16 — Upper bound on s
There is no (55, 68, large)-net in base 16, because
- 11 times m-reduction [i] would yield (55, 57, large)-net in base 16, but