Best Known (59, 72, s)-Nets in Base 16
(59, 72, 175020)-Net over F16 — Constructive and digital
Digital (59, 72, 175020)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (5, 11, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(5,256) in PG(10,16)) for nets [i] based on digital (0, 6, 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(5,256) in PG(10,16)) for nets [i] based on digital (0, 6, 257)-net over F256, 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 (5, 11, 257)-net over F16, using
(59, 72, 349549)-Net in Base 16 — Constructive
(59, 72, 349549)-net in base 16, 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
- (52, 65, 349525)-net in base 16, using
- net defined by OOA [i] based on OOA(1665, 349525, S16, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1665, 2097151, S16, 13), using
- discarding factors based on OA(1665, 2097155, S16, 13), using
- discarding parts of the base [i] based on linear OA(12837, 2097155, F128, 13) (dual of [2097155, 2097118, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(12837, 2097155, F128, 13) (dual of [2097155, 2097118, 14]-code), using
- discarding factors based on OA(1665, 2097155, S16, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1665, 2097151, S16, 13), using
- net defined by OOA [i] based on OOA(1665, 349525, S16, 13, 13), using
- digital (1, 7, 24)-net over F16, using
(59, 72, 5915487)-Net over F16 — Digital
Digital (59, 72, 5915487)-net over F16, using
(59, 72, large)-Net in Base 16 — Upper bound on s
There is no (59, 72, large)-net in base 16, because
- 11 times m-reduction [i] would yield (59, 61, large)-net in base 16, but