Best Known (210−15, 210, s)-Nets in Base 4
(210−15, 210, 4798950)-Net over F4 — Constructive and digital
Digital (195, 210, 4798950)-net over F4, using
- 41 times duplication [i] based on digital (194, 209, 4798950)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (30, 37, 5466)-net over F4, using
- net defined by OOA [i] based on linear OOA(437, 5466, F4, 7, 7) (dual of [(5466, 7), 38225, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(437, 16399, F4, 7) (dual of [16399, 16362, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [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(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(437, 16399, F4, 7) (dual of [16399, 16362, 8]-code), using
- net defined by OOA [i] based on linear OOA(437, 5466, F4, 7, 7) (dual of [(5466, 7), 38225, 8]-NRT-code), 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
- digital (30, 37, 5466)-net over F4, using
- (u, u+v)-construction [i] based on
(210−15, 210, large)-Net over F4 — Digital
Digital (195, 210, large)-net over F4, using
- 7 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
(210−15, 210, large)-Net in Base 4 — Upper bound on s
There is no (195, 210, large)-net in base 4, because
- 13 times m-reduction [i] would yield (195, 197, large)-net in base 4, but