Best Known (69, 69+15, s)-Nets in Base 16
(69, 69+15, 150054)-Net over F16 — Constructive and digital
Digital (69, 84, 150054)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 13, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(6,256) in PG(12,16)) for nets [i] based on digital (0, 7, 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(6,256) in PG(12,16)) for nets [i] based on digital (0, 7, 257)-net over F256, using
- digital (56, 71, 149797)-net over F16, using
- net defined by OOA [i] based on linear OOA(1671, 149797, F16, 15, 15) (dual of [(149797, 15), 2246884, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1671, 1048580, F16, 15) (dual of [1048580, 1048509, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(1671, 1048581, F16, 15) (dual of [1048581, 1048510, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(1671, 1048576, F16, 15) (dual of [1048576, 1048505, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(1666, 1048576, F16, 14) (dual of [1048576, 1048510, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(1671, 1048581, F16, 15) (dual of [1048581, 1048510, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1671, 1048580, F16, 15) (dual of [1048580, 1048509, 16]-code), using
- net defined by OOA [i] based on linear OOA(1671, 149797, F16, 15, 15) (dual of [(149797, 15), 2246884, 16]-NRT-code), using
- digital (6, 13, 257)-net over F16, using
(69, 69+15, 299617)-Net in Base 16 — Constructive
(69, 84, 299617)-net in base 16, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 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
- (61, 76, 299593)-net in base 16, using
- net defined by OOA [i] based on OOA(1676, 299593, S16, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(1676, 2097152, S16, 15), using
- discarding factors based on OA(1676, 2097155, S16, 15), using
- discarding parts of the base [i] based on linear OA(12843, 2097155, F128, 15) (dual of [2097155, 2097112, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(14) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(12843, 2097155, F128, 15) (dual of [2097155, 2097112, 16]-code), using
- discarding factors based on OA(1676, 2097155, S16, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(1676, 2097152, S16, 15), using
- net defined by OOA [i] based on OOA(1676, 299593, S16, 15, 15), using
- digital (1, 8, 24)-net over F16, using
(69, 69+15, 6762181)-Net over F16 — Digital
Digital (69, 84, 6762181)-net over F16, using
(69, 69+15, large)-Net in Base 16 — Upper bound on s
There is no (69, 84, large)-net in base 16, because
- 13 times m-reduction [i] would yield (69, 71, large)-net in base 16, but