Best Known (209−19, 209, s)-Nets in Base 4
(209−19, 209, 1864132)-Net over F4 — Constructive and digital
Digital (190, 209, 1864132)-net over F4, using
- 43 times duplication [i] based on digital (187, 206, 1864132)-net over F4, using
- trace code for nets [i] based on digital (84, 103, 932066)-net over F16, using
- net defined by OOA [i] based on linear OOA(16103, 932066, F16, 19, 19) (dual of [(932066, 19), 17709151, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(16103, 8388595, F16, 19) (dual of [8388595, 8388492, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(16103, large, F16, 19) (dual of [large, large−103, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(16103, large, F16, 19) (dual of [large, large−103, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(16103, 8388595, F16, 19) (dual of [8388595, 8388492, 20]-code), using
- net defined by OOA [i] based on linear OOA(16103, 932066, F16, 19, 19) (dual of [(932066, 19), 17709151, 20]-NRT-code), using
- trace code for nets [i] based on digital (84, 103, 932066)-net over F16, using
(209−19, 209, large)-Net over F4 — Digital
Digital (190, 209, large)-net over F4, using
- 42 times duplication [i] based on digital (188, 207, large)-net over F4, using
- t-expansion [i] based on digital (186, 207, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4207, large, F4, 21) (dual of [large, large−207, 22]-code), using
- 26 times code embedding in larger space [i] based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 26 times code embedding in larger space [i] based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4207, large, F4, 21) (dual of [large, large−207, 22]-code), using
- t-expansion [i] based on digital (186, 207, large)-net over F4, using
(209−19, 209, large)-Net in Base 4 — Upper bound on s
There is no (190, 209, large)-net in base 4, because
- 17 times m-reduction [i] would yield (190, 192, large)-net in base 4, but