Best Known (48−9, 48, s)-Nets in Base 16
(48−9, 48, 262402)-Net over F16 — Constructive and digital
Digital (39, 48, 262402)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 7, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(3,256) in PG(6,16)) for nets [i] based on digital (0, 4, 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(3,256) in PG(6,16)) for nets [i] based on digital (0, 4, 257)-net over F256, using
- digital (32, 41, 262145)-net over F16, using
- net defined by OOA [i] based on linear OOA(1641, 262145, F16, 9, 9) (dual of [(262145, 9), 2359264, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(1641, 1048581, F16, 9) (dual of [1048581, 1048540, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(1641, 1048576, F16, 9) (dual of [1048576, 1048535, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(1636, 1048576, F16, 8) (dual of [1048576, 1048540, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(1641, 1048581, F16, 9) (dual of [1048581, 1048540, 10]-code), using
- net defined by OOA [i] based on linear OOA(1641, 262145, F16, 9, 9) (dual of [(262145, 9), 2359264, 10]-NRT-code), using
- digital (3, 7, 257)-net over F16, using
(48−9, 48, 524305)-Net in Base 16 — Constructive
(39, 48, 524305)-net in base 16, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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
- (35, 44, 524288)-net in base 16, using
- net defined by OOA [i] based on OOA(1644, 524288, S16, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(1644, 2097153, S16, 9), using
- discarding factors based on OA(1644, 2097155, S16, 9), using
- discarding parts of the base [i] based on linear OA(12825, 2097155, F128, 9) (dual of [2097155, 2097130, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(12825, 2097152, F128, 9) (dual of [2097152, 2097127, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- discarding parts of the base [i] based on linear OA(12825, 2097155, F128, 9) (dual of [2097155, 2097130, 10]-code), using
- discarding factors based on OA(1644, 2097155, S16, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(1644, 2097153, S16, 9), using
- net defined by OOA [i] based on OOA(1644, 524288, S16, 9, 9), using
- digital (0, 4, 17)-net over F16, using
(48−9, 48, 4210359)-Net over F16 — Digital
Digital (39, 48, 4210359)-net over F16, using
(48−9, 48, large)-Net in Base 16 — Upper bound on s
There is no (39, 48, large)-net in base 16, because
- 7 times m-reduction [i] would yield (39, 41, large)-net in base 16, but