Best Known (192−15, 192, s)-Nets in Base 4
(192−15, 192, 4793564)-Net over F4 — Constructive and digital
Digital (177, 192, 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 (157, 172, 4793484)-net over F4, using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- digital (13, 20, 80)-net over F4, using
(192−15, 192, 4793572)-Net in Base 4 — Constructive
(177, 192, 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 (157, 172, 4793484)-net over F4, using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- (13, 20, 88)-net in base 4, using
(192−15, 192, large)-Net over F4 — Digital
Digital (177, 192, large)-net over F4, using
- 5 times m-reduction [i] based on digital (177, 197, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4197, large, F4, 20) (dual of [large, large−197, 21]-code), using
- 17 times code embedding in larger space [i] based on linear OA(4180, large, F4, 20) (dual of [large, large−180, 21]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 17 times code embedding in larger space [i] based on linear OA(4180, large, F4, 20) (dual of [large, large−180, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4197, large, F4, 20) (dual of [large, large−197, 21]-code), using
(192−15, 192, large)-Net in Base 4 — Upper bound on s
There is no (177, 192, large)-net in base 4, because
- 13 times m-reduction [i] would yield (177, 179, large)-net in base 4, but