Best Known (22, 31, s)-Nets in Base 4
(22, 31, 256)-Net over F4 — Constructive and digital
Digital (22, 31, 256)-net over F4, using
- net defined by OOA [i] based on linear OOA(431, 256, F4, 9, 9) (dual of [(256, 9), 2273, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(431, 1025, F4, 9) (dual of [1025, 994, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(431, 1025, F4, 9) (dual of [1025, 994, 10]-code), using
(22, 31, 512)-Net over F4 — Digital
Digital (22, 31, 512)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(431, 512, F4, 2, 9) (dual of [(512, 2), 993, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(431, 1024, F4, 9) (dual of [1024, 993, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 2-folding [i] based on linear OA(431, 1024, F4, 9) (dual of [1024, 993, 10]-code), using
(22, 31, 24173)-Net in Base 4 — Upper bound on s
There is no (22, 31, 24174)-net in base 4, because
- 1 times m-reduction [i] would yield (22, 30, 24174)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 153111 846515 487939 > 430 [i]