Best Known (211−15, 211, s)-Nets in Base 4
(211−15, 211, 4798952)-Net over F4 — Constructive and digital
Digital (196, 211, 4798952)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (32, 39, 5468)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (29, 36, 5463)-net over F4, using
- net defined by OOA [i] based on linear OOA(436, 5463, F4, 7, 7) (dual of [(5463, 7), 38205, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(436, 16390, F4, 7) (dual of [16390, 16354, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(436, 16391, F4, 7) (dual of [16391, 16355, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(429, 16384, F4, 6) (dual of [16384, 16355, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(436, 16391, F4, 7) (dual of [16391, 16355, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(436, 16390, F4, 7) (dual of [16390, 16354, 8]-code), using
- net defined by OOA [i] based on linear OOA(436, 5463, F4, 7, 7) (dual of [(5463, 7), 38205, 8]-NRT-code), using
- digital (0, 3, 5)-net over F4, using
- (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 (32, 39, 5468)-net over F4, using
(211−15, 211, large)-Net over F4 — Digital
Digital (196, 211, large)-net over F4, using
- t-expansion [i] based on digital (195, 211, large)-net over F4, using
- 6 times m-reduction [i] based on digital (195, 217, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4217, large, F4, 22) (dual of [large, large−217, 23]-code), using
- strength reduction [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- strength reduction [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4217, large, F4, 22) (dual of [large, large−217, 23]-code), using
- 6 times m-reduction [i] based on digital (195, 217, large)-net over F4, using
(211−15, 211, large)-Net in Base 4 — Upper bound on s
There is no (196, 211, large)-net in base 4, because
- 13 times m-reduction [i] would yield (196, 198, large)-net in base 4, but