Erasure Channels in Communication Theory

• Erasure channels are communication models where transmitted symbols may be erased with a known probability and the receiver is informed about each erasure.
• They underpin capacity-achieving codes and sharp threshold behaviors using techniques such as EXIT functions and doubly transitive code structures.
• They support diverse applications including channel polarization, quantum communications, caching, and distributed computing in unreliable environments.

An erasure channel is a canonical model for communication systems in which symbols transmitted over a channel may be erased with known probability, with the property that the receiver is explicitly informed of any erasure event. The erasure channel is a fundamental object in coding theory, information theory, and digital communications, serving as both a practical abstraction for bursty or packet-based loss and an analytically tractable model for developing code constructions and proving sharp threshold behaviors. The binary erasure channel (BEC), q-ary generalizations, erasure broadcast channels, erasure queue-channels, and quantum erasure channels are all extensively studied subtypes.

1. Formal Definition and Basic Properties

The binary erasure channel BEC($\epsilon$) maps each input bit $X \in \{0,1\}$ to an output $Y$ according to

$$Y = \begin{cases} X & \text{with probability } 1-\epsilon, \ * & \text{with probability } \epsilon, \end{cases}$$

where "$*$" denotes an erasure symbol known to the receiver (Kudekar et al., 2016). For codewords $\mathbf{X} \in \{0,1\}^N$, each coordinate is independently erased with probability $\epsilon$.

The capacity of the BEC is

$C = 1 - \epsilon,$

as shown by standard mutual information arguments (Kudekar et al., 2016). For q-ary input alphabets or more structured output, a generalized erasure channel is defined by an erasure probability on each symbol or a set of erasure probabilities mapping inputs to output cosets, recovering the BEC for $q=2$ (Sakai et al., 2016, Sakai et al., 2018).

2. Capacity-Achieving Codes and Sharp Thresholds

Capacity achieving codes for erasure channels are characterized by their ability to recover the original message with probability tending to one for channel erasure probabilities below $1 - R$, where $X \in \{0,1\}$0 is the code rate. The principal result for deterministic codes is:

Theorem (Capacity-achieving criterion): If $X \in \{0,1\}$1 is a sequence of binary linear codes of increasing blocklength with $X \in \{0,1\}$2 and the permutation group is doubly transitive, then under bit-MAP decoding over $X \in \{0,1\}$3, $X \in \{0,1\}$4 i.e., $X \in \{0,1\}$5 achieves capacity (Kudekar et al., 2016).

For Reed-Muller codes, with parameters chosen such that the code rate converges to $X \in \{0,1\}$6, the codes achieve BEC capacity under MAP decoding (Kudekar et al., 2016). The proof employs EXIT functions and the area theorem (integral of the average EXIT equals the code rate), as well as sharp threshold results for symmetric monotone Boolean functions, notably leveraging the Friedgut–Kalai–Bourgain theorem and doubly transitive group action (Kudekar et al., 2016).

Generalized bounds for block error in terms of subcode support weights (generalized Hamming weights $X \in \{0,1\}$7) relate the block error threshold to the bit error threshold (Pfister et al., 10 Jan 2025). In codes such as Reed-Muller, for which the sequence of minimal subcode supports grows sufficiently fast, block and bit error thresholds coincide up to $X \in \{0,1\}$8, tightly characterizing reliability as the channel erasure probability varies (Pfister et al., 10 Jan 2025).

3. Polarization and Multilevel Generalizations

Polar coding for erasure channels exploits the recursive self-similarity and tractable conditional entropy structure. For the BEC, under channel polarization, the synthetic channels produced remain BECs with recursively updated erasure probabilities. The construction generalizes to q-ary alphabets and multilevel "generalized erasure channels" where the output alphabet reveals partial information (e.g., $X \in \{0,1\}$9), and the polar transform acts via explicit recursions (Sakai et al., 2016, Sakai et al., 2018, Sakai et al., 2018): $Y$0 In the prime-power case, polarization occurs to $Y$1 levels, and the fraction of polarized channels at each level equals the initial erasure vector coefficients $Y$2 (Sakai et al., 2016, Sakai et al., 2018). For composite input sizes, multilevel channel polarization yields limiting proportions of partially noiseless channels that can be computed explicitly via convergent sequences and combinatorial recursions, as demonstrated for modular arithmetic erasure channels (Sakai et al., 2018).

4. Erasure Channels in Network and Broadcast Models

The erasure broadcast channel (EBC) and its caching variants are models where multiple receivers experience independent erasures per transmission. Optimal coding strategies exploit both channel state feedback and side information, e.g., through decentralized cache placement or piggybacking private messages for strong receivers onto messages for weaker receivers. Capacity-memory-rate trade-offs are precisely characterized, e.g.,

$Y$3

for symmetric EBCs with cache and feedback, where $Y$4 is user cache fraction and $Y$5 the erasure probability (Ghorbel et al., 2016). Joint cache–channel coding, particularly with asymmetric cache allocations, strictly enlarges achievable regions versus separate designs (Timo et al., 2015).

The erasure queue-channel (EQC) generalizes erasure behavior to queueing systems with time-dependent and correlated erasures, e.g., due to sojourn-time-dependent degradation (as relevant for quantum systems). Capacity is determined by

$Y$6

where $Y$7 is the arrival rate and $Y$8 the stationary waiting time distribution. Coding for such channels can employ interleaving across renewal blocks to mimic memoryless behavior (Mandalapu et al., 2023).

5. Quantum and Continuous-Variable Erasure Channels

Quantum erasure channels (QEC) extend the erasure channel model to quantum states. For a $Y$9-dimensional system, the QEC with erasure probability $$Y = \begin{cases} X & \text{with probability } 1-\epsilon, \ * & \text{with probability } \epsilon, \end{cases}$$0 maps $$Y = \begin{cases} X & \text{with probability } 1-\epsilon, \ * & \text{with probability } \epsilon, \end{cases}$$1, erasing pure states into an orthogonal flag state (Bao et al., 2023). Hypercontractivity inequalities, crucial in classical coding and information theory, are generalized to the quantum setting for these channels via multipartite log-Sobolev inequalities. These results yield nearly tight bounds on classical communication complexity for quantum randomness generation assisted by noisy entanglement (Bao et al., 2023).

Continuous-variable (CV) erasure channels, relevant for infinite-dimensional systems (e.g., optical modes), are modeled by protecting against erasure with a two-level flag. By projecting onto energy-constrained effective subspaces, code constructions and quantum capacity formulas carry over from the discrete case: $$Y = \begin{cases} X & \text{with probability } 1-\epsilon, \ * & \text{with probability } \epsilon, \end{cases}$$2 with physical entropic constraints determined by mean photon number (Zhong et al., 2022).

6. Applications: Caching, Computation, and Information Freshness

Erasure channels underlie the analysis of status update ("age of information"), distributed computation, and multi-agent learning in unreliable or delayed systems:

• Age of Information: Optimal strategies (Last-Come First-Serve no-buffer) and coding (random or MDS block codes) are derived for minimizing time-average age over memoryless $$Y = \begin{cases} X & \text{with probability } 1-\epsilon, \ * & \text{with probability } \epsilon, \end{cases}$$3-ary erasure channels and their block-coded extensions, with closed-form and asymptotic expressions for achievable and optimal age (Najm et al., 2019).
• Coded Distributed Computing: Packet erasure channels model link failures in master–worker computation architectures. Use of $$Y = \begin{cases} X & \text{with probability } 1-\epsilon, \ * & \text{with probability } \epsilon, \end{cases}$$4 MDS-coding with probabilistic retransmission over erasure links yields sharp upper and lower latency bounds and guides redundancy selections for reliability versus bandwidth (Han et al., 2019).
• Multi-Agent Bandit Learning: Action erasure channels formalize loss or delay in command transmissions from a learner to multiple agents. Algorithms combining repetition protocols (persistently resend until with high probability at least one copy is delivered) and batch scheduling yield sublinear regret, overcoming naive approaches which suffer linear loss under erasures (Hanna et al., 2023).

7. Extensions, Limits, and Algorithmic Implications

Erasure channel models have led to progress in:

• Interactive Communication: Explicit, efficient coding schemes match proven impossibility-of-tolerance thresholds (1/2 for general alphabets, 1/3 for binary alphabet) for adversarial erasure noise in interactive protocols, achievable with constant-size alphabet and polynomial time (Efremenko et al., 2015).
• Wireless Fading as Erasure: Concatenated inner convolutional and outer polar codes, with block interleaving, degrade Rayleigh fading to a BEC model for the polar decoder. Design trade-offs in inner code rate, interleaving depth, and block length are quantified for achievable end-to-end rates and reliability (Usman, 2020).
• Open Problems: Quantifying bit-to-block error threshold gaps for general codes, extending combinatorial block-error bounds to other channels, finitelength analyses, proving capacity-achieving polarization for queue-dependent erasure dynamics, and developing efficient decoding algorithms for block-MAP and non-memoryless settings remain important directions (Pfister et al., 10 Jan 2025, Mandalapu et al., 2023).

Model/Setting	Capacity/Threshold	Key Coding/Structural Principle
BEC($$Y = \begin{cases} X & \text{with probability } 1-\epsilon, \ * & \text{with probability } \epsilon, \end{cases}$$5)	$$Y = \begin{cases} X & \text{with probability } 1-\epsilon, \ * & \text{with probability } \epsilon, \end{cases}$$6	Dbl. transitive codes; sharp threshold; EXIT
q-ary erasure / gen.	$$Y = \begin{cases} X & \text{with probability } 1-\epsilon, \ * & \text{with probability } \epsilon, \end{cases}$$7	Multilevel polar, group coset structure
Erasure broadcast/cache	See polyhedral region	Joint cache-channel, feedback, piggybacking
Queue-channel (EQC)	$$Y = \begin{cases} X & \text{with probability } 1-\epsilon, \ * & \text{with probability } \epsilon, \end{cases}$$8	Renewal theory, interleaving wrapper
Quantum erasure	$$Y = \begin{cases} X & \text{with probability } 1-\epsilon, \ * & \text{with probability } \epsilon, \end{cases}$$9	Log-Sobolev, hypercontractivity
CV quantum erasure	$*$0	Entropy-constrained random coding
Interactive/adversarial	Up to $*$1 (binary)	Parity tagging, error-detect. retry

The erasure channel thus serves as a cornerstone for fundamental results in coding theory, polarization, optimal real-time and distributed computation, quantum communications, and robust algorithm design (Kudekar et al., 2016, Sakai et al., 2016, Pfister et al., 10 Jan 2025, Bao et al., 2023, Timo et al., 2015, Han et al., 2019).