Best Known (165, 165+14, s)-Nets in Base 4
(165, 165+14, 4793555)-Net over F4 — Constructive and digital
Digital (165, 179, 4793555)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (12, 19, 71)-net over F4, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 17)-net over F4, using
- s-reduction based on digital (0, 1, s)-net over F4 with arbitrarily large s, using
- digital (1, 3, 17)-net over F4, using
- s-reduction based on digital (1, 3, 21)-net over F4, using
- digital (1, 4, 17)-net over F4, using
- net defined by OOA [i] based on linear OOA(44, 17, F4, 3, 3) (dual of [(17, 3), 47, 4]-NRT-code), using
- digital (4, 11, 20)-net over F4, using
- digital (0, 1, 17)-net over F4, using
- generalized (u, u+v)-construction [i] based on
- digital (146, 160, 4793484)-net over F4, 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 (12, 19, 71)-net over F4, using
(165, 165+14, large)-Net over F4 — Digital
Digital (165, 179, large)-net over F4, using
- 43 times duplication [i] based on digital (162, 176, large)-net over F4, using
- t-expansion [i] based on digital (158, 176, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4176, large, F4, 18) (dual of [large, large−176, 19]-code), using
- 19 times code embedding in larger space [i] based on linear OA(4157, large, F4, 18) (dual of [large, large−157, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 19 times code embedding in larger space [i] based on linear OA(4157, large, F4, 18) (dual of [large, large−157, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4176, large, F4, 18) (dual of [large, large−176, 19]-code), using
- t-expansion [i] based on digital (158, 176, large)-net over F4, using
(165, 165+14, large)-Net in Base 4 — Upper bound on s
There is no (165, 179, large)-net in base 4, because
- 12 times m-reduction [i] would yield (165, 167, large)-net in base 4, but