Best Known (90, 104, s)-Nets in Base 16
(90, 104, 2400854)-Net over F16 — Constructive and digital
Digital (90, 104, 2400854)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (17, 24, 4112)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 257)-net over F16, using
- s-reduction based on digital (0, 0, s)-net over F16 with arbitrarily large s, using
- digital (0, 0, 257)-net over F16 (see above)
- digital (0, 0, 257)-net over F16 (see above)
- digital (0, 0, 257)-net over F16 (see above)
- digital (0, 0, 257)-net over F16 (see above)
- digital (0, 0, 257)-net over F16 (see above)
- digital (0, 0, 257)-net over F16 (see above)
- digital (0, 0, 257)-net over F16 (see above)
- digital (0, 0, 257)-net over F16 (see above)
- digital (0, 1, 257)-net over F16, using
- s-reduction based on digital (0, 1, s)-net over F16 with arbitrarily large s, using
- digital (0, 1, 257)-net over F16 (see above)
- digital (0, 1, 257)-net over F16 (see above)
- digital (0, 1, 257)-net over F16 (see above)
- digital (1, 3, 257)-net over F16, using
- s-reduction based on digital (1, 3, 273)-net over F16, using
- digital (1, 4, 257)-net over F16, using
- net defined by OOA [i] based on linear OOA(164, 257, F16, 3, 3) (dual of [(257, 3), 767, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(164, 257, F16, 2, 3) (dual of [(257, 2), 510, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(164, 257, F16, 3, 3) (dual of [(257, 3), 767, 4]-NRT-code), using
- 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 (0, 0, 257)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (66, 80, 2396742)-net over F16, using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- digital (17, 24, 4112)-net over F16, using
(90, 104, 2407665)-Net in Base 16 — Constructive
(90, 104, 2407665)-net in base 16, using
- (u, u+v)-construction [i] based on
- (17, 24, 10923)-net in base 16, using
- net defined by OOA [i] based on OOA(1624, 10923, S16, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(1624, 32770, S16, 7), using
- discarding factors based on OA(1624, 32771, S16, 7), using
- discarding parts of the base [i] based on linear OA(3219, 32771, F32, 7) (dual of [32771, 32752, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(3219, 32768, F32, 7) (dual of [32768, 32749, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(3216, 32768, F32, 6) (dual of [32768, 32752, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(320, 3, F32, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(3219, 32771, F32, 7) (dual of [32771, 32752, 8]-code), using
- discarding factors based on OA(1624, 32771, S16, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(1624, 32770, S16, 7), using
- net defined by OOA [i] based on OOA(1624, 10923, S16, 7, 7), using
- digital (66, 80, 2396742)-net over F16, using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- (17, 24, 10923)-net in base 16, using
(90, 104, large)-Net over F16 — Digital
Digital (90, 104, large)-net over F16, using
- t-expansion [i] based on digital (89, 104, large)-net over F16, using
- 5 times m-reduction [i] based on digital (89, 109, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16109, large, F16, 20) (dual of [large, large−109, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16109, large, F16, 20) (dual of [large, large−109, 21]-code), using
- 5 times m-reduction [i] based on digital (89, 109, large)-net over F16, using
(90, 104, large)-Net in Base 16 — Upper bound on s
There is no (90, 104, large)-net in base 16, because
- 12 times m-reduction [i] would yield (90, 92, large)-net in base 16, but