Best Known (49, 60, s)-Nets in Base 16
(49, 60, 209973)-Net over F16 — Constructive and digital
Digital (49, 60, 209973)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (4, 9, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(4,256) in PG(8,16)) for nets [i] based on digital (0, 5, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base reduction for projective spaces (embedding PG(4,256) in PG(8,16)) for nets [i] based on digital (0, 5, 257)-net over F256, using
- digital (40, 51, 209716)-net over F16, using
- net defined by OOA [i] based on linear OOA(1651, 209716, F16, 11, 11) (dual of [(209716, 11), 2306825, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(1651, 1048581, F16, 11) (dual of [1048581, 1048530, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(1651, 1048576, F16, 11) (dual of [1048576, 1048525, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(1646, 1048576, F16, 10) (dual of [1048576, 1048530, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(1651, 1048581, F16, 11) (dual of [1048581, 1048530, 12]-code), using
- net defined by OOA [i] based on linear OOA(1651, 209716, F16, 11, 11) (dual of [(209716, 11), 2306825, 12]-NRT-code), using
- digital (4, 9, 257)-net over F16, using
(49, 60, 419447)-Net in Base 16 — Constructive
(49, 60, 419447)-net in base 16, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- (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
- net defined by OOA [i] based on OOA(1655, 419430, S16, 11, 11), using
- digital (0, 5, 17)-net over F16, using
(49, 60, 5065302)-Net over F16 — Digital
Digital (49, 60, 5065302)-net over F16, using
(49, 60, large)-Net in Base 16 — Upper bound on s
There is no (49, 60, large)-net in base 16, because
- 9 times m-reduction [i] would yield (49, 51, large)-net in base 16, but