Best Known (227, 227+22, s)-Nets in Base 4
(227, 227+22, 1525200)-Net over F4 — Constructive and digital
Digital (227, 249, 1525200)-net over F4, using
- 47 times duplication [i] based on digital (220, 242, 1525200)-net over F4, using
- trace code for nets [i] based on digital (99, 121, 762600)-net over F16, using
- net defined by OOA [i] based on linear OOA(16121, 762600, F16, 22, 22) (dual of [(762600, 22), 16777079, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(16121, 8388600, F16, 22) (dual of [8388600, 8388479, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(16121, large, F16, 22) (dual of [large, large−121, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(16121, large, F16, 22) (dual of [large, large−121, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(16121, 8388600, F16, 22) (dual of [8388600, 8388479, 23]-code), using
- net defined by OOA [i] based on linear OOA(16121, 762600, F16, 22, 22) (dual of [(762600, 22), 16777079, 23]-NRT-code), using
- trace code for nets [i] based on digital (99, 121, 762600)-net over F16, using
(227, 227+22, large)-Net over F4 — Digital
Digital (227, 249, large)-net over F4, using
- 42 times duplication [i] based on digital (225, 247, large)-net over F4, using
- t-expansion [i] based on digital (222, 247, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4247, large, F4, 25) (dual of [large, large−247, 26]-code), using
- 30 times code embedding in larger space [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]
- 30 times code embedding in larger space [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(4247, large, F4, 25) (dual of [large, large−247, 26]-code), using
- t-expansion [i] based on digital (222, 247, large)-net over F4, using
(227, 227+22, large)-Net in Base 4 — Upper bound on s
There is no (227, 249, large)-net in base 4, because
- 20 times m-reduction [i] would yield (227, 229, large)-net in base 4, but