Decentralized Pliable Index Coding

• Decentralized Pliable Index Coding (DPIC) is a framework enabling users to broadcast encoded messages so each can decode an arbitrary subset of unknown messages under pliable and security constraints.
• It employs peer-to-peer transmission models and linear covering schemes, such as one-shot and 3-sum-cover, to optimize transmissions even under heterogeneous side-information conditions.
• DPIC’s application in federated learning and wireless data exchange demonstrates its potential to reduce communication overhead while maintaining privacy and enhancing network efficiency.

Decentralized Pliable Index Coding (DPIC) generalizes classical index coding to networks without a central server, with users (or clients) broadcasting messages in a peer-to-peer fashion under the pliable decoding paradigm. In DPIC, each user can be satisfied by decoding any subset of messages it does not already know, often subject to additional security constraints. This framework has been developed to address communication efficiency, privacy, and practicality in distributed systems ranging from wireless data exchange to federated learning.

1. Mathematical Model and Problem Formulation

DPIC considers a system of $m$ messages $\mathcal{M} = \{w_1, \ldots, w_m\}$ and $m$ users $\mathcal{U} = \{u_1, \ldots, u_m\}$ (Liu et al., 2020, Liu et al., 2020). Each user $u_j$ possesses a side-information set $A_j \subseteq \mathcal{M}$ (possibly determined via a specific pattern, such as $s$-circular shift), and wishes to decode any $t$ messages not in $A_j$. Communication occurs via a shared, error-free broadcast channel, with no central server.

• Transmission Model: In time slot $t$, a user $u_j$ may transmit $$x_j = \mathsf{ENC}_j(A_j, \mathcal{A}) \in \mathbb{F}_q^{\ell_j \kappa}$$, a function of its side information and (potentially) the overall side-information profile $\mathcal{A}$.
• Decoding Requirement: Each user must be able to decode (any) $t$ messages $w_{d_j^{(1)}}, ..., w_{d_j^{(t)}}$ with $d_j^{(k)} \notin A_j$, possibly subject to security constraints that prevent learning of any extra information (Liu et al., 2020, Liu et al., 2019).

Generalizations include (a) clients with heterogeneous side-information cardinality and target demand sizes, handled via linearly progressive, fixed-overlap (LPS–FO) side-information models (Padmanabhan et al., 3 Feb 2026), and (b) satisfaction via recovering a specified number of new messages, as in $t$-pliable DPIC or CDPIC$(S, K)$ (Kadakkottiri et al., 1 Jul 2025, Liu et al., 2019).

2. Performance Metrics and Converse Bounds

The fundamental metric is the total number of transmissions $T$ required for all users to achieve their targets, under the constraint that each transmission is a linear combination over a sufficiently large field $\mathbb{F}_q$ (linear DPIC) or possibly a general function (information-theoretic DPIC):

• Worst-case $T$: The minimum $T$ required to satisfy any instance within the specified family (e.g., all $s$-shifted instances for given $m, s$).
• Information-theoretic lower bounds: For homogeneous side-information, $T^* \geq m/s$ (when $m/(m-s) \in \mathbb{Z}$), and $T^* \geq 3m/(2s)$ (when $m/(m-s) \notin \mathbb{Z}$ for linear DPIC) (Liu et al., 2020, Liu et al., 2020).
• Heterogeneous targets and side-information: With $C$ clients and linearly growing side-information (LPS–FO), strict “exact-$T$” security requires $N(C) = C + N(C - r_{\max})$ transmissions, where $r_{\max}$ depends on the current number of active clients (Padmanabhan et al., 3 Feb 2026).
• Multiplicity Gap: For secure decentralized DPIC with $s$-circular-shift side-information, the cost of decentralization (relative to the centralized secure case) can incur a multiplicative gap up to $3$ in transmission length, whereas it is at most $2$ in the non-secure case (Liu et al., 2020, Liu et al., 2020).

3. Achievable Coding Schemes and Construction

Achievability results in DPIC rely on the structure of the side information and on whether security is required.

Linear Covering Schemes

• Homogeneous circular-shift (one-shot covering): For $m/(m-s) \in \mathbb{Z}$, users are grouped into contiguous blocks of size $m-s$. Within each block, users transmit the XOR (or sum) of their missing messages, ensuring each user decodes exactly one new message, achieving $T = m/s$ (Liu et al., 2020, Liu et al., 2020).
• 3-sum-cover schemes: For $m/(2s) \in \mathbb{Z}$, in each group of $2s$ users, three transmissions suffice such that each user recovers exactly one new message, yielding $T = 3m/(2s)$ (Liu et al., 2020).
• LPS–FO recursion for heterogeneous targets: For linearly progressive, fixed-overlap side-information, a recursive scheme partitions users into $r_{\max}$-blocks and accomplishes “exact-$T$” targeting with $N(C) = C + N(C - r_{\max})$ transmissions. Each client only learns the requisite number of new messages, and others remain unaffected in each recursion (Padmanabhan et al., 3 Feb 2026).

Optimality and Patch Constructions

• For $C$ clients with side-information window $K$ (CDPIC$(S, K)$), exact conditions for optimality are established via explicit code constructions: uncoded message transmissions for small $K$, pairwise XORs for larger $K$, and multi-way sums in the dense regime. Optimal numbers of broadcasts are known for several parameter regimes (Kadakkottiri et al., 1 Jul 2025).

4. Security Constraints and Proof Techniques

Security in DPIC tightens the requirement: users must not be able to learn more than their target number of messages.

• Information-theoretic security: $$I\bigl(\mathcal{M} \setminus (A_j \cup \{w_{d_j}\});\, \mathbf{x},\, A_j\bigr) = 0$$ for each user $u_j$, ensuring zero leakage about other messages (Liu et al., 2020, Liu et al., 2020).
• Converse arguments: The “chain-of-pairs” argument analyzes the structure of coding vectors to bound how many users can be satisfied by a given transmission while preserving security (lower bounding $T$) (Liu et al., 2020).
• Security in recursion: In LPS–FO schemes, the overlap structure ensures that only active clients can decode information in each recursion level, and all other clients see only XORs of unknown messages (Padmanabhan et al., 3 Feb 2026).

Infeasible parameter regimes are also characterized; for instance, no secure decentralized linear DPIC exists for $(s=1, m \geq 3)$, $(s=2, m \geq 5)$, $(s=3, m$ odd), or $(s = m-2, m$ odd) (Liu et al., 2020).

5. Applications and Empirical Evaluations

DPIC has been applied as the foundational primitive in distributed, privacy-sensitive systems.

• Federated Learning (FL) data shuffling: Consecutive DPIC (CDPIC$(S, K)$) protocols are used for efficient data shuffling among edge devices (e.g., RSUs in ITS), enhancing convergence and accuracy of FL under non-IID data. Broadcast-efficient DPIC schemes reduce the number of required transmissions by up to $60\%$, lower latency, and minimize gradient exchanges (Kadakkottiri et al., 1 Jul 2025).
• Empirical performance: On MNIST/CIFAR-10 with $C=10,K=6,7$, the optimal use of DPIC codes achieves substantial throughput gains and rapid accuracy improvement—e.g., FedAvg accuracy increases from $91\%$ to $98.9\%$ with only 4 shuffling rounds, and FL round count drops from 80 to 10 (Kadakkottiri et al., 1 Jul 2025).

$S$ (new classes)	$N_{\text{uncoded}}$	$N_{\text{CDPIC}}$	$\text{FedAvg acc.}@5\text{ rounds}$
0	–	–	91.0\%
3	8	4	98.9\%

6. Comparison with Centralized and Non-Secure Models

In the classical (centralized) PICOD, a single transmitter exploits global knowledge and can produce scalar random linear combinations. In contrast, DPIC must respect each user’s local encoding constraints.

• Capacity equivalence: For the complete-$S$ class with $s_{\min}\neq m-t$, decentralized and centralized PICOD have identical optimal code length. However, the codes themselves differ: decentralized DPIC relies on sparse MDS or message-splitting vector-linear codes to satisfy local-encoding requirements (Liu et al., 2019).
• Security multiplicative gap: The price of achieving strong security in a decentralized setting is quantifiably higher: up to a factor $3$ increase in minimum code length versus the centralized secure PICOD counterpart, and strict “exact-$T$” security with heterogeneity introduces additive overheads $N(C - r_{\max})$ not present in homogeneous or non-secure models (Liu et al., 2020, Liu et al., 2020, Padmanabhan et al., 3 Feb 2026).

7. Open Problems and Future Directions

Several avenues remain for DPIC research:

• Nonlinear codes and stronger security: Current infeasibility results pertain to linear codes. Extending these to general codes or considering block-security and collusion-resistant models is an open challenge (Padmanabhan et al., 3 Feb 2026).
• Side-information graphs: Extending DPIC results to more general, non-circular side-information patterns (e.g., arbitrary or circular-arc graphs) is identified as a key direction (Liu et al., 2020).
• Tightness and adaptivity: Reducing or characterizing the additive penalty $N(C - r_{\max})$ in the LPS–FO model and developing adaptive or field-size-optimal constructions remain important.
• Practical deployments: Further empirical validation is required for deployment in edge networks, federated learning, and other distributed systems with privacy and efficiency constraints.

DPIC thus serves as both a theoretically rich and practically relevant framework for distributed coded communication under pliable, secure, and resource-aware constraints.