Information Theory and Channel Coding

Introduction and Context

Wireless communication systems operate over channels affected by thermal noise, interference, fading, shadowing, multipath propagation, and hardware imperfections. These impairments can cause transmitted bits to be received incorrectly. Information theory establishes the fundamental limits of reliable communication, while channel coding provides practical methods for approaching those limits.

The central question is: how much information can be transmitted reliably through a noisy channel, and what signal-processing methods are required to achieve that reliability? Claude Shannon’s framework answers the first question through entropy, mutual information, and channel capacity. Coding theory addresses the second through structured redundancy, decoding algorithms, and retransmission strategies.

Modern cellular, satellite, optical, and wireless systems combine source representation, modulation, channel coding, interleaving, link adaptation, and hybrid automatic repeat request (HARQ). Convolutional codes remain important in legacy and low-complexity systems, whereas turbo, low-density parity-check (LDPC), and polar codes are central to contemporary standards. Fountain codes are useful when packet loss is substantial or feedback is unavailable.

Learning Objectives

After studying this chapter, the learner should be able to:

1. Define entropy, conditional entropy, mutual information, and channel capacity.

2. Apply Shannon’s channel-capacity limits to bandwidth- and power-limited links.

3. Explain the distinction between coding rate, code rate, spectral efficiency, and signal-to-noise ratio.

4. Describe convolutional, turbo, LDPC, polar, fountain, and hybrid-ARQ techniques.

5. Calculate coding gain and interpret its effect on link performance.

6. Explain how link adaptation selects modulation and coding schemes.

7. Evaluate practical coding systems, including their complexity, latency, and finite-blocklength limitations.

8. Analyze an applied wireless-link design using coding and adaptation principles.

1. Information Measures

1.1 Information and uncertainty

If an event has probability p, its self-information is

\[

I(x)=-\log_2 p(x) \quad \text{bits}.

\]

A rare event conveys more information than a predictable event. The logarithm base 2 expresses information in bits. For a discrete random variable X with probability mass function p(x), the entropy is

\[

H(X)=-\sum_x p(x)\log_2 p(x).

\]

Entropy is the average uncertainty of X. It is maximized by a uniform distribution. For a binary variable with probability P(X=1)=p,

\[

H_2(p)=-p\log_2 p-(1-p)\log_2(1-p).

\]

The maximum binary entropy is 1 bit, obtained when p=0.5. Entropy is not necessarily the number of bits used by a practical representation; efficient source coding attempts to approach the entropy, whereas channel coding deliberately adds redundancy.

1.2 Conditional entropy and mutual information

Conditional entropy measures residual uncertainty about X when Y is known:

\[

H(X|Y)=-\sum_{x,y}p(x,y)\log_2 p(x|y).

\]

Mutual information measures the reduction in uncertainty about X obtained from observing Y:

\[

I(X;Y)=H(X)-H(X|Y).

\]

Equivalent forms are

\[

I(X;Y)=H(Y)-H(Y|X)=H(X)+H(Y)-H(X,Y).

\]

Mutual information is nonnegative and equals zero when X and Y are statistically independent. In a communication system, X is the transmitted symbol and Y is the received observation. Mutual information therefore quantifies the information conveyed by the channel for a specified input distribution.

1.3 Data-processing principle

If X→Y→Z forms a Markov chain, meaning that Z depends on X only through Y, then

\[

I(X;Z)\leq I(X;Y).

\]

This data-processing inequality states that processing cannot create information about the original input. Practical demodulation, quantization, and decoding may preserve or lose information, but cannot exceed the information available in the channel observation.

2. Channel Capacity and Shannon Limits

2.1 Discrete memoryless channels

A discrete memoryless channel (DMC) is characterized by transition probabilities p(y|x), with each channel use depending only on the current input and not on previous uses. Its capacity is

\[

C=\max_{p(x)} I(X;Y) \quad \text{bits per channel use}.

\]

The maximization is essential: capacity is the largest mutual information achievable by choosing the best input distribution, subject to channel constraints.

For a binary symmetric channel with crossover probability ε, the output bit is flipped independently with probability ε. With a uniform input distribution,

\[

C=1-H_2(\varepsilon) \quad \text{bits per channel use}.

\]

This result applies to an ideal memoryless binary channel and assumes independent errors.

2.2 Additive white Gaussian noise channel

For a band-limited additive white Gaussian noise (AWGN) channel with bandwidth B hertz and received average signal power S, the noise has two-sided power spectral density N0/2 and total noise power N=N0B. Shannon capacity is

\[

C=B\log_2\left(1+\frac{S}{N}\right)

=B\log_2\left(1+\frac{S}{N_0B}\right) \quad \text{bits/s}.

\]

The formula assumes a stationary AWGN channel, an average-power constraint, sufficiently long codes, and reliable decoding with arbitrarily small error probability. It is a fundamental limit, not a guaranteed performance of a particular code.

The ratio S/N is the received signal-to-noise ratio (SNR). Spectral efficiency is

\[

\eta=\frac{C}{B}=\log_2(1+\mathrm{SNR}) \quad \text{bits/s/Hz}

\]

at the Shannon limit. In a practical system, the useful information rate is lower because of coding overhead, pilots, guard intervals, control signaling, finite blocklength, implementation loss, and fading margins.

2.3 Energy per bit and the wideband limit

Let R be the information rate in bits per second. The received energy per information bit is

\[

E_b=\frac{S}{R}.

\]

Using R=C and defining γ=E_b/N0, the AWGN capacity relation can be written as

\[

\frac{C}{B}=\log_2\left(1+\frac{E_b}{N_0}\frac{C}{B}\right).

\]

As spectral efficiency approaches zero, the theoretical minimum required energy per bit approaches

\[

\left(\frac{E_b}{N_0}\right)_{\min}=\ln 2\approx -1.59\ \text{dB}.

\]

This is an asymptotic limit requiring vanishing spectral efficiency and very long, ideal codes. It is not a practical operating target for ordinary high-throughput links.

Worked Example 1: AWGN capacity

A link has bandwidth B=1 MHz and received SNR=15 dB. First convert SNR to linear units:

\[

\mathrm{SNR}=10^{15/10}=31.62.

\]

Therefore,

\[

C=10^6\log_2(1+31.62)

=10^6\log_2(32.62)

\approx 5.03\ \text{Mbit/s}.

\]

This is the ideal asymptotic capacity. A practical coded throughput must be lower.

3. Channel Coding Fundamentals

3.1 Purpose of coding

A channel encoder maps k information bits into n transmitted bits or symbols. The code rate is

\[

R_c=\frac{k}{n}, \qquad 0<R_c\leq 1.

\]

The n−k additional bits are redundancy. A lower code rate generally improves error protection but reduces the fraction of transmitted bits carrying user information. Coding does not increase Shannon capacity; it enables a system to operate closer to capacity at a specified error probability.

The effective information rate for a modulation carrying m=log2(M) bits per symbol is approximately

\[

R_b=R_s\,m\,R_c,

\]

where Rs is the symbol rate. This expression assumes that all symbols are used for data and excludes pilots, control information, cyclic prefixes, and other overhead. The net rate must account for those factors separately.

3.2 Minimum distance and error protection

For a block code, the minimum Hamming distance dmin is the smallest Hamming distance between any two distinct valid codewords. A code can detect up to dmin−1 errors and can correct up to

\[

t=\left\lfloor\frac{d_{\min}-1}{2}\right\rfloor

\]

errors under nearest-neighbor decoding. These are worst-case guarantees and do not directly predict the error rate under random noise.

For soft-decision decoding, the receiver retains reliability information, such as log-likelihood ratios (LLRs), instead of making an immediate binary decision. Soft information generally provides a significant performance advantage over hard decisions, at the cost of memory and computational complexity.

3.3 Coding gain

Coding gain is commonly defined as the reduction in required SNR, often expressed in decibels, to achieve the same error-rate target compared with an uncoded reference. If the uncoded and coded required SNRs are γu and γc, respectively,

\[

G_c=10\log_{10}\left(\frac{\gamma_u}{\gamma_c}\right)\ \text{dB}.

\]

The definition is meaningful only when the comparison uses the same modulation, bandwidth, information rate, channel model, block length, decoder assumptions, and error-rate target. A nominal coding gain quoted without these conditions can be misleading.

4. Major Coding Families

4.1 Convolutional codes

A convolutional encoder generates output bits from current and previous input bits using shift registers and modulo-2 adders. Its memory introduces a trellis structure. Important parameters include the input length k, output length n, code rate k/n, constraint length, and generator polynomials.

The Viterbi algorithm performs maximum-likelihood sequence detection on the trellis. It uses an additive path metric and retains the best path entering each state. Its complexity grows with the number of trellis states and the decoded sequence length. The BCJR algorithm computes symbol-by-symbol a posteriori probabilities and is useful when soft outputs are required.

Convolutional codes offer predictable performance and relatively mature implementations. Their limitations include increasing complexity with constraint length, sequential decoding latency, and weaker asymptotic performance than modern capacity-approaching block codes at long block lengths.

4.2 Turbo codes

Turbo coding uses two or more recursive systematic convolutional encoders connected through an interleaver. The component decoders exchange extrinsic information iteratively. This iterative process can approach the Shannon limit for sufficiently long blocks and appropriate interleavers.

Turbo codes exhibit an error floor at sufficiently low error rates, caused by harmful low-weight codewords and finite interleaver effects. They also incur iterative decoding latency and energy consumption. Turbo codes were widely used in earlier cellular standards, including 3G and 4G systems.

4.3 LDPC codes

An LDPC code is defined by a sparse parity-check matrix H. A valid codeword c satisfies

\[

Hc^T=0

\]

over the binary field GF(2). The Tanner graph represents variable nodes, corresponding to code bits, and check nodes, corresponding to parity constraints.

Belief propagation, also called sum-product decoding, exchanges probabilistic messages between these nodes. The min-sum algorithm is a lower-complexity approximation. LDPC codes can provide excellent performance with parallel hardware, making them suitable for high-throughput systems such as Wi-Fi, 5G data channels, and satellite links.

Important design issues include block length, degree distribution, girth, puncturing, rate matching, decoder iterations, quantization, and memory bandwidth. Short cycles in the Tanner graph can reduce the independence of messages and degrade iterative decoding.

4.4 Polar codes

Polar codes use channel polarization: through a recursive transform, synthetic bit channels become either highly reliable or highly unreliable as block length increases. Information bits are placed in reliable bit channels, while frozen bits are placed in unreliable channels.

Successive-cancellation (SC) decoding processes bits sequentially. Successive-cancellation list (SCL) decoding retains several candidate paths and can approach maximum-likelihood performance for short blocks. A cyclic redundancy check can be concatenated with polar coding to select the most plausible candidate.

Polar codes are especially attractive for control information and short-to-moderate block lengths. Their sequential dependencies may limit throughput unless specialized parallel or hardware-optimized architectures are used.

4.5 Fountain codes

Fountain codes are rateless codes: the encoder can generate a potentially unlimited sequence of encoded packets, and the receiver collects packets until successful recovery is possible. LT and Raptor codes are important examples.

For an erasure channel, fountain coding is valuable because the transmitter need not know in advance which packets will be lost. The receiver typically needs slightly more than k encoded packets to recover k source packets, with the overhead depending on the code design and decoding failure probability.

Fountain codes are useful in multicast, broadcast, deep-space communication, content distribution, and networks with variable packet loss. They are less naturally suited to channels dominated by soft Gaussian errors unless combined with modulation and inner error-correcting codes.

5. Hybrid ARQ and Error-Control Strategies

Automatic repeat request (ARQ) detects errors and requests retransmission. Forward error correction (FEC) attempts to correct errors without retransmission. Hybrid ARQ combines both.

In Type-I HARQ, each retransmission repeats the complete coded packet. In incremental-redundancy HARQ, the first transmission contains only part of a mother codeword; later transmissions provide additional parity. The receiver combines all observations and decodes a lower-rate effective code. Chase combining repeats the same coded bits and combines their received energy, often improving SNR approximately according to the accumulated observations when noise conditions are independent.

HARQ improves reliability and spectral efficiency under variable channel conditions, but it introduces feedback overhead, variable latency, buffer requirements, and possible packet reordering. It is unsuitable for applications with extremely strict delay bounds unless carefully configured.

A practical protocol must define the maximum number of transmissions, acknowledgment timing, soft-buffer size, packet segmentation, combining method, and behavior when the deadline expires.

6. Link Adaptation

Link adaptation selects modulation, coding rate, transmit power, resource allocation, or repetition level according to channel conditions. The receiver estimates channel quality using pilot symbols, reference signals, or decoded acknowledgment statistics. It reports a channel-quality indicator (CQI), while the transmitter selects a modulation and coding scheme (MCS).

Higher-order modulation and high code rates provide greater throughput when SNR and channel reliability are high. Robust modulation and lower code rates are selected under weak or rapidly varying conditions. The objective is not simply to maximize instantaneous data rate; it is usually to maximize expected goodput subject to error-rate, latency, fairness, and power constraints.

A simplified goodput model is

\[

G=R(1-P_e),

\]

where R is the attempted information rate and Pe is the packet error probability. This model ignores retransmission timing and protocol overhead, so system designers often use measured or simulated throughput tables instead.

Link adaptation must account for channel-estimation error, feedback delay, Doppler spread, interference, and correlation between the reported CQI and the channel at transmission time. A conservative margin reduces packet failures but may waste capacity; an aggressive margin increases throughput in favorable conditions but can cause retransmission bursts.

7. Finite Blocklength and Fading Channels

Shannon capacity assumes arbitrarily long codewords. Wireless packets are finite, and short control messages may contain only tens or hundreds of bits. At finite blocklength, a nonzero error probability must be accepted even when the rate is below asymptotic capacity. A useful normal approximation for many regular channels is

\[

R(n,\varepsilon)\approx C-\sqrt{\frac{V}{n}}Q^{-1}(\varepsilon)+\frac{\log_2 n}{2n},

\]

where n is blocklength in channel uses, ε is target block error probability, V is channel dispersion in squared bits per channel use, and Q−1 is the inverse Gaussian Q-function. The approximation is not universal and should be applied only under the channel model and blocklength conditions for which it is validated.

Fading makes capacity dependent on the channel state. With perfect channel knowledge at the receiver, a block-fading channel may be modeled by an instantaneous SNR γ. Coding across sufficiently many independent fades can average the channel, whereas short packets may experience outage. Outage probability is the probability that the instantaneous channel cannot support the selected transmission rate. Diversity, interleaving, adaptive modulation, HARQ, and retransmission combining reduce the impact of fading.

8. Worked Design Example: Coded Wireless Link

Consider an illustrative 1 MHz link using 16-QAM, which carries 4 coded bits per symbol. The selected code rate is Rc=3/4 and the symbol rate is 800 ksymbol/s. Assume 10% of resources are consumed by pilots, control, and guard overhead.

The gross coded bit rate is

\[

R_{gross}=800,000\times4=3.2\ \text{Mbit/s}.

\]

The information rate before non-payload protocol losses is

\[

R_{info}=3.2\times\frac{3}{4}=2.4\ \text{Mbit/s}.

\]

After 10% overhead, the net payload rate is

\[

R_{net}=2.4\times0.90=2.16\ \text{Mbit/s}.

\]

The nominal spectral efficiency relative to the 1 MHz occupied bandwidth is 2.16 bit/s/Hz. This value must be compared with the actual operating SNR and target error probability, not merely with the modulation order. If the measured packet error rate becomes excessive, the system may select QPSK with a lower code rate, add incremental redundancy, or reduce the scheduled rate.

9. Applied Case Study: Adaptive Industrial Wireless Sensor Link

Background

The following is an illustrative case. A factory uses wireless sensors to report vibration and temperature measurements to an edge controller. Metal structures create multipath fading, and moving machinery intermittently blocks the line of sight. Packets are short, and control messages have a deadline of 20 ms.

Problem

A fixed high-rate MCS provides good throughput near the access point but produces frequent packet failures near machinery. Repeated failures cause stale sensor information and excessive energy consumption. A fixed robust MCS avoids failures but reduces the number of sensors that can be supported.

Analysis

The channel varies over time, so average received power alone is not sufficient for selecting an MCS. Short packets also suffer finite-blocklength penalties. The system therefore needs channel-quality estimation, a small MCS table, bounded HARQ, and a deadline-aware scheduler.

Approach

The proposed design uses a short LDPC code with rate matching. The receiver computes soft LLRs and stores them for incremental-redundancy HARQ. A pilot-based CQI estimate is filtered to reduce rapid MCS oscillations. Three operating modes are defined:

• Robust mode: QPSK with a low code rate for poor channel conditions.
• Normal mode: QPSK or 16-QAM with an intermediate code rate.
• High-throughput mode: 16-QAM with a high code rate when channel quality is stable.

At most two retransmissions are permitted. If the packet deadline is close, the scheduler may transmit a fresh robust packet rather than waiting for another retransmission. Packets are prioritized by age as well as sensor importance.

Outcome

The design trades peak rate for improved deadline reliability. Incremental redundancy is more efficient than repeating the complete packet because successful decoding can use all accumulated parity. The bounded retransmission policy prevents a small number of difficult links from monopolizing channel resources.

Learning Points

1. Coding, modulation, scheduling, and retransmission must be designed jointly.

2. CQI delay and estimation error can make an apparently optimal MCS unreliable.

3. Goodput and deadline reliability are more useful objectives than raw physical-layer rate.

4. Soft combining requires memory and processing resources that must be included in the implementation budget.

5. Industrial systems should define behavior for missed deadlines rather than relying solely on eventual retransmission.

10. Design Trade-offs, Limitations, and Best Practices

No coding family is universally optimal. Convolutional codes are attractive for simple trellis-based implementations. Turbo codes offer strong iterative performance but can have latency and error-floor issues. LDPC codes support highly parallel high-throughput decoding but may require substantial memory bandwidth. Polar codes are well suited to certain short-block control channels but require careful list-decoder design. Fountain codes are effective for erasure and multicast environments but introduce recovery overhead and are not a replacement for all inner codes.

Best practices include:

• Compare codes at equal information rate, blocklength, bandwidth, and target error probability.
• Report whether SNR is measured per symbol, per coded bit, or per information bit.
• Include pilot, cyclic-prefix, control, and retransmission overhead in throughput calculations.
• Use soft decoding when the receiver can support the required quantization and memory.
• Validate link adaptation under channel-estimation error, feedback delay, and mobility.
• Evaluate latency distribution, not only average throughput.
• Consider decoder energy, thermal limits, and memory traffic in hardware design.
• Use realistic channel models, including interference and non-Gaussian disturbances.

11. Contemporary Developments and Broader Implications

Modern research combines coding with probabilistic shaping, coded modulation, machine-learning-assisted decoding, and semantic or task-oriented communication. Neural methods may assist channel estimation, decoder scheduling, or learned belief-propagation weights, but they must be evaluated for generalization, explainability, computational cost, and reliability outside the training distribution.

In 5G and emerging 6G systems, coding is increasingly integrated with flexible numerology, network slicing, distributed antennas, reconfigurable surfaces, and joint sensing and communication. These developments do not invalidate Shannon theory; rather, they introduce additional constraints such as finite latency, correlated interference, nonstationary channels, and energy limits.

Security also matters. Error-control redundancy can reveal structural information about traffic and may interact with encryption, authentication, and integrity protection. Coding should not be treated as a substitute for cryptographic security. Standards compliance, spectrum regulation, lawful-intercept obligations, electromagnetic compatibility, and dependable operation in safety-critical applications must be considered during system deployment.

Chapter Summary

Information theory quantifies uncertainty and the information conveyed through a noisy channel. Entropy measures average uncertainty, mutual information measures the information shared by transmitted and received variables, and channel capacity is the maximum mutual information under channel constraints. For the AWGN channel, capacity is B log2(1+S/N), but practical systems operate below this limit because of finite blocklength, fading, overhead, implementation loss, and nonideal decoding.

Channel coding adds structured redundancy so that errors can be detected and corrected. Convolutional, turbo, LDPC, polar, and fountain codes address different combinations of blocklength, throughput, latency, hardware complexity, and channel characteristics. HARQ combines forward error correction with retransmission, while link adaptation selects modulation and coding according to channel quality. Effective wireless design requires joint consideration of coding gain, goodput, reliability, latency, energy, feedback, and implementation constraints.

Key Terms

Entropy: Average uncertainty of a random variable, measured in bits.

Mutual information: Information shared between transmitted and received variables.

Channel capacity: Maximum reliable information rate permitted by a channel model and its constraints.

AWGN: Additive white Gaussian noise channel model.

Spectral efficiency: Information rate per unit bandwidth.

Code rate: Ratio of information symbols or bits to transmitted coded symbols or bits.

Redundancy: Additional information introduced to support error detection or correction.

Minimum distance: Smallest distance between distinct codewords.

Soft decision: Decoding input that retains reliability information.

Coding gain: Required-SNR reduction obtained by coding at a specified target.

Convolutional code: Code generated using shift-register memory and a trellis.

Turbo code: Iterative code based on concatenated recursive convolutional encoders.

LDPC code: Code defined by a sparse parity-check matrix.

Polar code: Code based on channel polarization and frozen bit positions.

Fountain code: Rateless code that generates encoded packets until recovery is possible.

HARQ: Hybrid automatic repeat request combining coding and retransmission.

Incremental redundancy: HARQ method that sends additional parity on retransmission.

Link adaptation: Dynamic selection of transmission parameters based on channel conditions.

Finite blocklength: Regime in which codewords are short and asymptotic capacity is insufficient to predict performance.

Channel dispersion: Parameter describing finite-blocklength variation around capacity.

Review Questions

Short Answer Questions

1. Define entropy and explain why it is maximized by a uniform distribution.

2. What is the difference between code rate and spectral efficiency?

3. State the Shannon capacity formula for an AWGN channel and define its variables.

4. Why do soft-decision decoders generally outperform hard-decision decoders?

5. Distinguish Type-I HARQ from incremental-redundancy HARQ.

Long Answer Questions

1. Derive the relationship between AWGN capacity, spectral efficiency, and Eb/N0, and explain the significance of the wideband limit.

2. Compare convolutional, turbo, LDPC, and polar codes in terms of decoding method, latency, complexity, and typical application.

3. Explain how fading and finite blocklength alter the interpretation of Shannon capacity in wireless systems.

4. Discuss the design factors that must be included when quoting coding gain.

5. Explain the operation of link adaptation and analyze the effects of CQI error and feedback delay.

Analytical / Application Questions

1. A 2 MHz AWGN link operates at an SNR of 10 dB. Calculate its ideal Shannon capacity and explain why a practical throughput would be lower.

2. A system uses 64-QAM, code rate 2/3, and a symbol rate of 1 Msymbol/s. Calculate the gross coded bit rate and information bit rate before protocol overhead. Discuss two reasons why the net rate will be lower.

3. Design an error-control strategy for a mobile telemetry link with rapidly varying fading, 10 ms latency, and a target packet error probability below 1%. Justify your choices of code family, MCS adaptation, and HARQ policy.

Multiple Choice Questions with Answers

Question 1

For a binary symmetric channel with crossover probability ε, assuming a uniform input, which expression gives capacity?

A. H2(ε)

B. 1−H2(ε)

C. 1+H2(ε)

D. εH2(ε)

Correct Answer: B. 1−H2(ε)

Question 2

A channel has bandwidth 1 MHz and SNR 15 dB. Its ideal AWGN capacity is approximately:

A. 1.00 Mbit/s

B. 3.16 Mbit/s

C. 5.03 Mbit/s

D. 15.00 Mbit/s

Correct Answer: C. 5.03 Mbit/s

Question 3

A code maps 600 information bits into 900 transmitted bits. What is its code rate?

A. 1/3

B. 2/3

C. 3/2

D. 900

Correct Answer: B. 2/3

Question 4

Which code uses a sparse parity-check matrix and a Tanner graph?

A. Convolutional code

B. Turbo code

C. LDPC code

D. Fountain code

Correct Answer: C. LDPC code

Question 5

Which technique sends additional parity from a mother code on retransmission?

A. Type-I ARQ

B. Incremental-redundancy HARQ

C. Source coding

D. Hard-decision demodulation

Correct Answer: B. Incremental-redundancy HARQ

Question 6

A modulation scheme carries 4 coded bits per symbol, uses code rate 3/4, and has a symbol rate of 800 ksymbol/s. Ignoring overhead, the information rate is:

A. 0.60 Mbit/s

B. 1.80 Mbit/s

C. 2.40 Mbit/s

D. 3.20 Mbit/s

Correct Answer: C. 2.40 Mbit/s

Question 7

What does mutual information I(X;Y) quantify in a communication system?

A. The bandwidth occupied by X

B. The reduction in uncertainty about X from observing Y

C. The number of parity bits in a code

D. The average transmit power

Correct Answer: B. The reduction in uncertainty about X from observing Y

Question 8

Which coding approach is particularly associated with channel polarization and frozen bits?

A. Polar coding

B. Convolutional coding

C. Chase combining

D. LT erasure coding only

Correct Answer: A. Polar coding

Question 9

For a code with minimum distance dmin=7, the guaranteed correction capability under nearest-neighbor decoding is:

A. 1 error

B. 2 errors

C. 3 errors

D. 6 errors

Correct Answer: C. 3 errors

Question 10

Why can a practical coded link not normally achieve the asymptotic Shannon capacity exactly?

A. Capacity applies only to uncoded systems

B. Practical systems have finite blocks, overhead, fading, and implementation loss

C. Coding always reduces received signal power to zero

D. Capacity is defined only for optical channels

Correct Answer: B. Practical systems have finite blocks, overhead, fading, and implementation loss