We also present the simple decoder architecture to decode the original data.
For group testing, the simple decoder for the classical model is shown to be more efficient than the one of Chan et al. and it provably achieves the simple group testing capacity.
The possibility of such an achievement was foreseen in [20], where it was shown that the simple decoder capacity becomes equal to the joint decoder capacity as c0 goes to infinity.
We will prove that asymptotically, the proposed universal joint decoder is equivalent to the universal simple decoder of Section 3, thus also achieving the joint uninformed capacity.
In case y=0, the combined simple decoder score of this tuple T (using the simple universal decoder g of Section 3) would be: begin{array}{*{20}l} sum_{j in T} S_{j,i} = z cdot g(1,0,p) + (c - z) g 0,0,p) = z lnleft(1 - frac{1}{c}right) + (c - z) lnleft(1 + frac{p}{c 1-p }right) sim -frac{z}{c 1-p }right - z)p}{c(1 - p)} = frac{p - z/c}{1 - p}. end{array}.
In this paper we analyze the performance of the capacity-achieving simple decoder of [21] in the Restricted Digit Model: Following the approach of [22], we use Bernstein's inequality and Bennett's inequality to upper bound the false-positive and false-negative error probability, respectively.
Finally, although it is hard to estimate the scores of mixed tuples with this decoder, just like in [37], we expect that the joint decoder score for a tuple is roughly equal to the sum of the c individual simple decoder scores.
Now, this almost fits the score-based simple decoder framework, except for that the terms inside the logarithm are not independent for different positions i.
So the joint universal decoder score for a tuple T is asymptotically equivalent to the sum of the simple universal decoder scores for the members in this tuple, if p∈[δ,1−δ].
This means that the simple universal decoder of Theorem 3 already asymptotically achieves the joint capacity.
