Best Known (180−14, 180, s)-Nets in Base 4
(180−14, 180, 4793564)-Net over F4 — Constructive and digital
Digital (166, 180, 4793564)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (13, 20, 80)-net over F4, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 20)-net over F4, using
- s-reduction based on digital (0, 1, s)-net over F4 with arbitrarily large s, using
- digital (1, 3, 20)-net over F4, using
- s-reduction based on digital (1, 3, 21)-net over F4, using
- digital (2, 5, 20)-net over F4, using
- s-reduction based on digital (2, 5, 41)-net over F4, using
- net defined by OOA [i] based on linear OOA(45, 41, F4, 3, 3) (dual of [(41, 3), 118, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(45, 41, F4, 2, 3) (dual of [(41, 2), 77, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(45, 41, F4, 3, 3) (dual of [(41, 3), 118, 4]-NRT-code), using
- s-reduction based on digital (2, 5, 41)-net over F4, using
- digital (4, 11, 20)-net over F4, using
- digital (0, 1, 20)-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 (13, 20, 80)-net over F4, using
(180−14, 180, 4793572)-Net in Base 4 — Constructive
(166, 180, 4793572)-net in base 4, using
- (u, u+v)-construction [i] based on
- (13, 20, 88)-net in base 4, using
- trace code for nets [i] based on (3, 10, 44)-net in base 16, using
- base change [i] based on digital (1, 8, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- base change [i] based on digital (1, 8, 44)-net over F32, using
- trace code for nets [i] based on (3, 10, 44)-net in base 16, using
- 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
- (13, 20, 88)-net in base 4, using
(180−14, 180, large)-Net over F4 — Digital
Digital (166, 180, large)-net over F4, using
- 44 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
(180−14, 180, large)-Net in Base 4 — Upper bound on s
There is no (166, 180, large)-net in base 4, because
- 12 times m-reduction [i] would yield (166, 168, large)-net in base 4, but