Multiple-Antenna Placement Delivery Array (MAPDA)

• MAPDA is a combinatorial framework that systematically encodes cache placement and one-shot delivery strategies in multi-antenna (MISO) systems.
• The design achieves near-optimal sum-DoF and significantly reduces subpacketization compared to classical coded caching methods through precise structural constraints.
• MAPDA unifies classical PDA concepts with modern multi-antenna techniques, offering flexible constructions applicable to partial connectivity, nonuniform caching, and multi-user retrieval scenarios.

A multiple-antenna placement delivery array (MAPDA) is a combinatorial structure fundamental to the design of coded caching schemes in multiple-input, single-output (MISO) broadcast systems and related multiterminal information networks. The MAPDA formalism generalizes the placement delivery array (PDA) concept introduced for single-antenna shared-link coded caching and provides a unified framework for array-based cache assignment and one-shot zero-forcing delivery in systems with $L \ge 1$ transmit antennas. MAPDA schemes enable uncoded placement and one-shot linear delivery, achieving near-optimal or optimal sum degrees of freedom (DoF) with dramatically reduced subpacketization relative to classic combinatorial coded caching methods. Recent research demonstrates MAPDA's versatility for diverse network architectures, including partial connectivity, multi-user information retrieval, and nonuniform scenarios.

1. Formal Definition and Key Structural Properties

An $(L,K,F,Z,S)$ MAPDA is an array $\mathbf{P}$ of size $F \times K$ whose entries are either a special symbol “$*$” or an integer in $\{1,2,\dots,S\}$. The formal definition is as follows (Namboodiri et al., 2022, Yang et al., 2022, Zheng et al., 15 Jan 2026):

• C1 (Column star-count): Each column has exactly $Z$ stars.
• C2 (Integer occurrence): Every integer $s \in \{1,\dots,S\}$ appears at least once in $\mathbf{P}$.
• C3 (Column integer uniqueness): No integer appears more than once in any column.
• C4 (Row integer multiplicity): For every integer $s$, consider the subarray $\mathbf{P}^{(s)}$ of rows and columns where $s$ appears; in each row of $\mathbf{P}^{(s)}$, the number of integer entries is at most $L$.

This structure encodes both the cache placement (indicated by stars) and the one-shot multicast delivery phase enabled by $L$-antenna transmitters (realized via zero-forcing). When every integer in $\mathbf{P}$ appears exactly $g$ times, $\mathbf{P}$ is called $g$-regular.

Connection to Classical PDAs

For $L=1$, the MAPDA structure reduces to the classical placement delivery array employed in single-antenna shared-link coded caching, where the key restriction is that each multicast transmission must be decodable by all its intended recipients through cached side information.

2. MAPDA Construction, Schemes, and Delivery Process

A MAPDA is instantiated for a system with $K$ users, $L$ transmit antennas, a file library of $N$ files, and per-user cache size $M$. Each file is divided into $F$ subfiles. The entries of the array dictate cache placement and delivery organization:

• Placement: User $k$ caches subfile $j$ of every file iff $P(j,k)= *$. This leads to per-user memory fraction $M/N=Z/F$.
• Delivery: For each integer $s$, the set of array positions $(j_1,k_1),\dots,(j_{g_s},k_{g_s})$ where $P(j_i,k_i)=s$ defines which users are served simultaneously in transmission $s$. The server forms an $L\times g_s$ precoder to broadcast those packets, zero-forcing interference in accordance with C4.
• Subpacketization: $F$ (the number of array rows) directly determines the required subpacketization.
• Delivery time and DoF: $T=S/F$; the achieved sum-DoF is $g=\dfrac{K(F-Z)}{S}$ (Namboodiri et al., 2022, Yang et al., 2022, Wang et al., 21 Jan 2025).

3. Achievable Degrees of Freedom and Optimality

In a $g$-regular MAPDA, the maximum sum-DoF is achieved when the scheme serves $g \le t+L$ users per transmission, where $t=KM/N\in\mathbb{Z}$ is the aggregate cache gain parameter. The MAPDA framework saturates the uncoded placement, one-shot linear delivery DoF bound: $\mathrm{DoF}_{\max} = t+L.$ For $L=1$, this coincides with the fundamental caching bound for shared-link coded caching. With MAPDAs, this optimal DoF is achievable under the zero-forcing constraints imposed by the physical channel and the combinatorial structure (i.e., the row-multiplicity bound in C4) (Namboodiri et al., 2022, Yang et al., 2022, Zheng et al., 15 Jan 2026).

4. Combinatorial Constructions and Subpacketization Analysis

MAPDA research has produced several explicit constructions, each targeting different regimes and parameters:

Construction	Regime	Achieved DoF	Subpacketization $F$
Latin-square MAPDA	$K = t+L$	$K$	$F=K$
$g$-regular “lifting”	General $(K,L,t)$	$L+m(g-1)$	$F = \alpha \binom{K/m}{t/m}$
EPDA (Construction I/II)	$K=t+L$ or $K=nt+(n-1)L$	$t+L$	$F=K/\gcd(K,t,L)$
HSDP Mapda (Zheng et al., 15 Jan 2026)	$F=K$ for linear $K$	close to $t+L$	$F=K$

MAPDA constructions reduce subpacketization requirements from previously exponential bounds, e.g., $F=\binom{K}{t}\binom{K-t-1}{L-1}$ (for SMK), to either linear ($F=K$) or subexponential in $K$, without substantial loss in delivery efficiency (Namboodiri et al., 2022, Yang et al., 2022, Wang et al., 21 Jan 2025, Zheng et al., 15 Jan 2026).

The HSDP (half-sum disjoint packing) technique (Zheng et al., 15 Jan 2026) synthesizes linear-subpacketization MAPDAs by embedding combinatorial subsets into the design, generalizing the Latin-square and orbit-based approaches, with the sum-DoF tradeoff converging rapidly to the optimal value for moderate $n$ and $q$ parameters.

5. Applications, Extensions, and Multi-Network Utility

The MAPDA paradigm now underpins a broad variety of coded caching applications:

• Cache-aided MISO BC: The canonical use case, with $L$ antennas and $K$ users, where MAPDAs enable uncoded placement and one-shot zero-forcing delivery, saturating the optimal DoF at low subpacketization (Yang et al., 2022, Namboodiri et al., 2022).
• Cache-aided MIMO and Partial Connectivity: MAPDA generalizations handle multiple antennas at each user (i.e., MIMO-PDA (Huang et al., 15 Jan 2026)), extensions to partially connected interference/topological networks (Cheng et al., 2023), and transmitter-side caching.
• Nonuniform/Location-aware Caching: Extensions such as LAPDA adapt MAPDA methodology to nonuniform user memory allocation, e.g., to mitigate delivery time for users in poor channel conditions (Mahmoodi et al., 2023).
• Multi-user Information Retrieval: In multi-user information retrieval systems, MAPDA-based schemes achieve optimal normalized delivery time (NDT) while reducing both computational complexity and subpacketization compared to prior art (Wang et al., 21 Jan 2025).
• Index Coding and Interference Networks: The MAPDA zero-forcing combinatorics naturally extend to index coding and general network coding scenarios with one-shot linear solutions.

6. Performance, Computational Complexity, and Comparisons

MAPDA-based schemes achieve dramatic gains in both performance and practicality. Most notably:

• Subpacketization: MAPDAs drive $F$ from exponential to linear or subexponential in $K$, e.g., $F=K/\gcd(K,t,L)$, with asymptotic improvements $F/F_\mathrm{prev} \to 0$ as $K \to \infty$ (Namboodiri et al., 2022, Wang et al., 21 Jan 2025).
• Delivery Time: Attain optimal $T = S/F = \frac{K(1-M/N)}{t+L}$ under uncoded placement and one-shot linear delivery (Yang et al., 2022).
• Computational Complexity: New MAPDA schemes, particularly those in (Wang et al., 21 Jan 2025), reduce the computational cost from exponential in classical solutions to $O((t+L)^3\,S)$ (polynomial in key parameters), critical for large-scale systems.
• Flexibility: The MAPDA formalism encapsulates all known uncoded-placement, one-shot MISO coded caching schemes, and enables a systematic approach for new designs via the combinatorial template (Zheng et al., 15 Jan 2026).

7. Theoretical Insights and Unifying Perspective

MAPDAs unify zero-forcing multicast design and coded caching subfile scheduling under a single array framework. The critical C4 property—controlling per-row integer multiplicity—precisely encodes the physical feasibility of simultaneous spatial transmission: each coded packet is transmitted only when the number of co-served users in a multicast group does not exceed the available spatial nulling degrees provided by $L$ antennas. Recursion, group algebra, and embedding techniques (such as HSDP) enable MAPDA constructions that map directly to the DoF limits of the network.

In summary, the MAPDA formalism consolidates and generalizes the array-theoretic approach to multi-antenna coded caching, exploits the spatial multiplexing and coded delivery interplay, and delivers both optimality in sum-DoF and practical implementability with dramatically minimized subpacketization (Yang et al., 2022, Namboodiri et al., 2022, Zheng et al., 15 Jan 2026, Wang et al., 21 Jan 2025, Cheng et al., 2023, Mahmoodi et al., 2023, Huang et al., 15 Jan 2026).