## 6 Conclusions  
[PERSON] et al. (2015); [PERSON] et al. (2016); [PERSON] et al. (2017) and [PERSON] et al. (2017) established foundational analyses of BER under single-frequency interference for long-code systems, but their models assume infinite PN sequences, making them inapplicable to short-period PN systems. Prior works like [PERSON] and [PERSON] (2015) and [PERSON] et al. (2018) investigated periodic sequence DSSS systems but lacked quantitative expressions for BER dependence on SIR, phase deviations, and frequency offsets. [PERSON] (2019) compared long- and short-code performance but did not address frequency-dependent phase shifts or threshold SIR effects. Advancements of this study:  
1. We derived a quantitative BER expression for short-code systems, explicitly linking BER to SIR, interference frequency offsets (e.g., \(\Delta f=kR_{\text{b}}\)), and phase deviations (\(\Delta\varphi\))--filling gaps in the literature ([PERSON] et al., 2016; [PERSON] et al., 2016; [PERSON] et al., 2016; [PERSON] et al., 2016; [PERSON] et al., 2016; [PERSON] et al., 2016; [PERSON] et al., 2016; [PERSON] et al., 2016; [PERSON] et al., 2016). 2. We demonstrated periodic BER oscillations under non-integer frequency offsets, validated experimentally (Figures 5-7), a phenomenon unaddressed in long-code analyses ([PERSON] et al., 2016; [PERSON] et al., 2016; [PERSON] et al., 2016). 3. We revealed the threshold SIR effect (e.g., \(-24\) dB for \(N=15\)) and its phase dependency (Figure 3), critical for practical hydroacoustic DSSS design. 4. We compared short- and long-code systems, showing divergent BER behaviors at \(\Delta f=0\) and non-integer multiples of Rb, advancing insights beyond ([PERSON] et al., 2016; [PERSON] et al., 2016). These contributions refine existing frameworks (e.g., ([PERSON] et al., 2016; [PERSON] et al., 2016)) by enabling rapid performance prediction for realistic underwater acoustic DSSS systems with short PN sequences. This expression holds importance in engineering practice. The implications of short-period systems' sensitivity to single-frequency interference can be analyzed through two application dimensions:  
1. Stealth Compromise in Military Systems  
Short-period systems (e.g., submarine tactical links or UUV swarm coordination) prioritize low-latency encrypted command transmission, but their vulnerability may:  
1. Increase interception risks: Adversaries could detect communication bands via targeted narrowband jamming. 2. Demand dynamic countermeasures: Requires real-time period switching using our theoretical model to balance speed and stealth. 2. Reliability-Cost Tradeoff in Civilian Deployments  
For civilian applications like ocean monitoring:  
1. Deployment constraints: Short-period sensors should avoid common interference bands. 2. Hybrid network solutions: Heterogeneous architectures combining long-period backbone nodes (monthly duty cycles) with short-period mobile units (hourly updates) achieve both timeliness and robustness. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sequence** & **Code** & **Primitive** & **Carrier** & **Bit Rate** & **SIR** & \(E_{b}\)-\(N_{0}\)**-1** \\
**No.** & **Length** & **Polynomial** & **Frequency/Hz** & **/bit-s\({}^{-1}\)** & **/dB** & **/dB** \\ \hline
1 & 31 & \(\begin{array}{c}1+x+x^{3}+\\ x^{4}+x^{5}\end{array}\) & 1240 & 5 & \(-20\) & 8.4 \\
2 & 31 & \(1+x^{3}+x^{5}\) & 1240 & 5 & \(-20\) & 8.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameter of Figure 7.  
**Funding:** This research received no external funding. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The raw data supporting the conclusions of this article will be made available by the authors on request. **Acknowledgments:** We acknowledge the equal contribution of all the authors. We thank the editors and the reviewers for their useful comments that improved this paper. **Conflicts of Interest:** The authors declare no conflicts of interest.