# Exploring ISAC: Information-Theoretic Insights
## 7 ISAC with Detection-Error Exponents
### Results on the Block Model  
We first focus on the model where coding at the Tx is restricted to be non-adaptive. Combining the results in [71; 72; 73; 74; 57], we obtain the following theorem:  
**Theorem 14**.: _Under non-adaptive coding, we have the following information-theoretic results for the rate-detection-exponent regions in the Stein setup, the symmetric setup, and in the exponents region sense._  
1. _In the_ Stein setup_, a non-negative rate-detection-error pair_ \((\mathsf{R},\mathsf{D}^{\rm nad})\) _is achievable if, and only if,_\[\mathsf{R} \leq \min_{\theta\in\Theta}I_{P_{Y|X}^{\theta}}(X;Y), \tag{83}\] \[\mathsf{D}^{\mathrm{nad}} \leq \sum_{x}P_{X}(x)D\Big{(}P_{Z|X}^{0}(\cdot|x)\big{\|}P_{Z|X}^{1}( \cdot|x) \tag{84}\]
2. _In the_ exponent-region sense_, a non-negative rate-detection-error pair \((\mathsf{R},\mathsf{D}^{\mathrm{nad}})\) is achievable if, and only if, for some input distribution \(P_{X}\): \[\mathsf{R} \leq \min_{\theta}I_{P_{Y|X}^{\theta}}(X;Y)\] (85) \[\mathsf{D}_{1}^{\mathrm{nad}} \leq \min_{P_{Z|X}:\atop\mathbb{E}_{P_{X}}\big{[}D(\tilde{P}_{Z|X}\|P_ {Z|X}^{0})\big{]}\leq\mathsf{D}_{0}^{\mathrm{nad}}}\mathbb{E}_{P_{X}}\Big{[}D \Big{(}\tilde{P}_{Z|X}\big{\|}P_{Z|X}^{1}\Big{)}\Big{]}\] (86)
3. _In the_ symmetric setup_, a non-negative rate-detection-exponent pair \((\mathsf{R},\mathsf{D}^{\mathrm{nad}})\) is achievable if, and only if, for some input distribution \(P_{X}\): \[\mathsf{R}\ \leq\ \min_{\theta\in\Theta}I_{P_{Y|X}^{\theta}}(X;Y),\] (87) \[\mathsf{D}^{\mathrm{nad}}\ \leq\ \min_{\theta\in\Theta}\min_{a^{ \prime}\in\Theta\setminus\theta}\max_{I\in[0,1]}-\sum_{x}P_{X}(x)\log\left(\sum _{z}\Big{(}P_{Z|XS}^{\theta}(z|x)\Big{)}^{l}\Big{(}P_{Z|XS}^{a}(z|x)\Big{)}^{ 1-l}\right).\] (88)  
As already mentioned, exactly characterizing the fundamental limits under adaptive coding seems a very challenging problem and for the moment only achievability results are known [73], which, however, prove the superiority of adaptive coding over non-adaptive coding. Note that it has been known for a long time that for the compound channel, adaptive coding increases communication rate because it allows the Tx to learn the hypothesis with high probability and then adapt the input distribution (and thus the code construction) to the actual transition law of the communication channel. This idea allows the following result to be obtained ([73], Theorem 5).  
**Theorem 15**.: _Under adaptive coding, a rate-detection-exponent pair \((\mathsf{R},\mathsf{D}^{\mathrm{nad}})\) is achievable in the symmetric setup if for any \(\theta\in\Theta\) there exists an input distribution \(P_{X}\) so that_  
\[\mathsf{R}\ \leq\ I_{P_{Y|X}^{\theta}}(X;Y), \tag{89}\]  
\[\mathsf{D}^{\mathrm{ad}}\ \leq\ \min_{a\
eq\theta}\max_{I\in[0,1]}-\sum_{x}P_{X}(x )\log\left(\sum_{z}\Big{(}P_{Z|X}^{\theta}(z|x)\Big{)}^{l}\Big{(}P_{Z|X}^{ \theta}(z|x)\Big{)}^{1-l}\right). \tag{90}\]  
As mentioned in [73], a further improved region can be achieved by using adaptive strategies also to improve the sensing parts, not only the communication parts. For a more detailed discussion, see [73].