Shared-Link Coded Caching via MAPDA

• Shared-Link Coded Caching Scheme is a strategy that leverages coded multicasting and optimized cache placement to reduce normalized delivery time in broadcast networks.
• It employs combinatorial designs like MAPDA to manage interference and subpacketization, ensuring practical complexity for multi-antenna and MIR systems.
• The approach extends classical MAN methods by integrating antenna constraints and grouping techniques to achieve near-optimal throughput with exponential subpacketization reductions.

A shared-link coded caching scheme is a data delivery strategy that exploits user-side caching and coded multicasting to reduce communication latency in networks where a set of users simultaneously request files from a shared content library. The framework is fundamentally based on careful cache placement and delivery strategies that are jointly designed to exploit coded-multicasting opportunities in the presence of a shared communication medium. In multi-antenna (MISO/MIMO) and multi-user information retrieval system (MIR) settings, these schemes leverage sophisticated combinatorial constructs—particularly the Multiple-Antenna Placement Delivery Array (MAPDA)—to achieve optimal delivery time and low subpacketization while managing interference using antenna resources (Wang et al., 21 Jan 2025, Yang et al., 2022, Namboodiri et al., 2022, Zheng et al., 15 Jan 2026).

1. Foundations of Shared-Link Coded Caching

The shared-link model considers a server (possibly with $L$ transmit antennas) connected via a broadcast channel to $K$ users, each equipped with a cache of size $M$. The server stores $N$ files; each user requests one file. The caching system operates in two phases:

• Placement phase: Users prefetch data into their caches, oblivious to future demands. Placement may be uncoded (direct copying of subpackets).
• Delivery phase: Upon revelation of user requests, the server transmits coded messages so that all demands can be satisfied.

A key goal is to minimize the normalized delivery time (NDT), equivalently maximizing sum Degrees of Freedom (DoF), with practical constraints on subpacketization (file splitting) and transmission complexity (Wang et al., 21 Jan 2025, Yang et al., 2022).

The canonical benchmark is the Maddah–Ali–Niesen (MAN) scheme, which achieves optimal memory–rate tradeoff in the single-antenna ($L=1$) broadcast channel. Extensions to multi-antenna and networked setups require new combinatorial design principles to leverage available spatial degrees of freedom.

2. The Multiple-Antenna Placement Delivery Array (MAPDA) Framework

MAPDA is a combinatorial object encoding both placement and delivery for multi-antenna coded caching in the shared-link model. Formally, an $(L,K,F,Z,S)$ MAPDA is an $F\times K$ array $\mathbf P$ whose entries are either the symbol $*$ (placement—representing cached subpackets) or an integer in $[S]$ (delivery—denoting coded transmission slots), and satisfies the following (Wang et al., 21 Jan 2025, Yang et al., 2022, Zheng et al., 15 Jan 2026):

• C1 (Placement constraint): Each column (user) contains exactly $Z$ stars; thus, user $k$ stores $Z/F$ fraction of each file.
• C2 (Coverage constraint): Every integer $s\in[S]$ appears at least once.
• C3 (Uniqueness): In any column, each integer $s$ appears at most once.
• C4 (One-shot decodability/interference constraint): For each $s$, let $\mathbf P^{(s)}$ be the subarray restricted to rows/columns where $s$ appears; in every row of $\mathbf P^{(s)}$, at most $L$ integer entries may appear, respecting the $L$-antenna constraint (i.e., up to $L$ simultaneous interferences can be handled in one transmission).

Given a MAPDA, the corresponding coded caching scheme operates as follows:

Phase	Mapping from MAPDA	Constraints
Placement	$*$ in column $k$, row $f$: cache	User $k$ stores subpacket $f$ of each file
Delivery	Integer $s$ in row $f$, col $k$: slot $s$ delivers subpacket $f$ of requested file to user $k$	At most $L$ integers in $\mathbf P^{(s)}$ per row

The performance metrics are:

• Subpacketization: $F$
• Memory ratio: $M/N = Z/F$
• Sum-DoF: $K(F-Z)/S$ (number of users per slot, time-normalized)
• Normalized Delivery Time: $\tau = S/F$

A central result establishes that, under uncoded placement and one-shot delivery, sum-DoF cannot exceed $L + K M/N$ (Yang et al., 2022, Namboodiri et al., 2022).

3. Construction Methodologies and Regimes

The literature provides several explicit MAPDA constructions for various regimes, optimizing for both subpacketization and DoF:

• EPDA-style (GCD-based) Construction: For $K, t=KM/N, L$ with $\alpha = \gcd(K, t, L)$ and $t+L \le K$, yields

$$F = \frac{t+L}{\alpha} \binom{K/\alpha}{(t+L)/\alpha},\quad Z = \frac{t}{\alpha} \binom{K/\alpha-1}{(t+L)/\alpha-1},\quad S = \frac{K-t}{\alpha} \binom{K/\alpha}{(t+L)/\alpha}$$

Provides dramatic subpacketization reduction compared to prior schemes (Wang et al., 21 Jan 2025, Namboodiri et al., 2022).

• YWCC-style (Grouping): Utilizes symmetries via group-based array construction; supports $m \mid K,\,m \le L$ (Yang et al., 2022).
• Latin Square-Based Linear-Subpacketization: For $(K = L + KM/N)$, builds $K \times K$ arrays with $F = K$ (Yang et al., 2022, Zheng et al., 15 Jan 2026); enables regimes with lowest subpacketization.
• Half-Sum Disjoint Packing (HSDP): Introduces a new combinatorial construction for achieving linear subpacketization via structured Latin squares, slightly reducing DoF but yielding $F = K$ and practical trade-off (Zheng et al., 15 Jan 2026).

For $K = t+L$, the canonical construction achieves both optimal DoF and minimal subpacketization $F=K$ (Namboodiri et al., 2022). More intricate regimes (e.g., $K=nt+(n-1)L$) are addressed by cyclic or grouping-based constructions. All constructions guarantee the critical antenna constraint (C4) ensuring one-shot decodability by leveraging the interference nulling capability of the available antennas.

4. Application to Multi-User Information Retrieval and Beyond

MAPDA-based shared-link coded caching generalizes directly to multi-user information retrieval (MIR) systems, where $K$ single-antenna users with cache connect via a base station (BS) with $L$ antennas that cannot directly access the content library. In this model, all beamforming is performed in the user uplink, with the BS acting as a forwarder (identity downlink) provided $KM/N \geq L$ (Wang et al., 21 Jan 2025). The delivery process is divided into:

• Uplink: Users in each served set encode linear combinations of missing subpackets, with beamforming vectors constructed to null out interference for uncached content.
• Downlink: The BS forwards the $L$-dimensional observation vector as is; each recipient decodes via cached-aided cancellation.

This arrangement achieves the information-theoretic optimum for NDT and sum-DoF, with the MAPDA reducing both subpacketization and computational complexity compared to traditional ASMST-type constructions (Wang et al., 21 Jan 2025).

MAPDA designs have been further extended to generalized network settings, including partially connected linear networks, multiple antenna receivers, and interference channels. Extensions accommodate additional system heterogeneity, non-uniform placement, or network topology-dependent delivery, while preserving combinatorial tractability (Cheng et al., 2023, Huang et al., 15 Jan 2026).

5. Subpacketization, Computational Complexity, and Performance Comparison

A defining strength of MAPDA-based approaches is the exponential improvement in subpacketization relative to classical MAN or combinatorial designs, particularly in the multi-antenna regime. For example (Wang et al., 21 Jan 2025, Yang et al., 2022, Namboodiri et al., 2022):

• Prior ASMST/MAN-based schemes: $F_{\rm ASMST}= \binom{K}{t} \binom{K-t-1}{L-1}$, growing doubly-exponentially in $K$ and $L$.
• EPDA/MAPDA-based schemes: $F = O(2^{K/\alpha})$ or $F=K$ in special regimes, subexponentially smaller.
• HSDP and Latin Square constructions: $F = K$, often with only a small DoF loss compared to optimal.

The computation of beamforming vectors is also greatly simplified—identity downlink in MIR settings, or $O((t+L)^3)$ per slot for one-shot delivery—versus the $O(\binom{t+L-1}{t}^3 \binom{K}{t+L})$ cost for ASMST.

A qualitative comparison is given below:

Scheme	Sum-DoF	Subpacketization $F$	Complexity per slot	Applicability
ASMST/MAN	$t+L$	$\binom{K}{t}\binom{K-t-1}{L-1}$	Exponential	General
EPDA/MAPDA	$t+L$ (optimal)	$O(2^{K/\alpha})$ or $K/\gcd(K,t,L)$	$O((t+L)^3)$	All regimes
HSDP/Latin Square	Near optimal	$K$	$O(K^3)$	$K=L+KM/N$ or reduced DoF

6. Extensions and Broader Implications

MAPDA structures unify the design of shared-link coded caching for a wide variety of systems, subsuming earlier constructs such as Placement Delivery Arrays (PDA, $L=1$ case), Extended PDA (EPDA), and providing a blueprint for systematic placement/delivery design in cache-aided wireless and network-coded settings (Yang et al., 2022, Namboodiri et al., 2022, Cheng et al., 2023, Huang et al., 15 Jan 2026).

Variants such as MIMO-PDA extend the concept to systems with multiple antennas at both transmitter and receiver, incorporating additional constraints into the placement and zero-forcing design (Huang et al., 15 Jan 2026). Extensions to location-aware, topology-sensitive, and partially connected networks further illustrate the broad applicability and combinatorial flexibility of MAPDA-like arrays (Mahmoodi et al., 2023, Cheng et al., 2023).

A plausible implication is that further reductions in subpacketization and complexity may be achievable by synthesizing alternative combinatorial frameworks (e.g., hybrid HSDP, Baranyai factorizations) or by relaxing DoF or placement symmetry constraints for improved scalability.

7. Concluding Synthesis

Shared-link coded caching under the MAPDA paradigm enables optimal or near-optimal normalized delivery time with low subpacketization and practical complexity in a range of multi-user, multi-antenna, and networked contexts. MAPDAs constitute a unifying combinatorial foundation that (a) simultaneously encodes placement and one-shot delivery, (b) adapts readily to antenna, cache, and topology constraints, and (c) supports both theoretical optimality and tractable implementation in large-scale cache-aided networks (Wang et al., 21 Jan 2025, Yang et al., 2022, Namboodiri et al., 2022, Cheng et al., 2023, Zheng et al., 15 Jan 2026).