EC-23.2 Block Codes: Hamming, Cyclic and Reed–Solomon
You can construct a linear block code from its generator matrix, decode by syndrome, explain why cyclic structure makes encoding a shift-register operation, and say what makes Reed–Solomon codes the standard answer to burst errors.
Before:S5-3. Digital Signal ProcessingEC-11. Data Communications and Computer Networks
A block code takes a fixed number of information symbols and produces a fixed number of transmitted ones. The whole family is linear algebra over a finite field, which sounds forbidding and turns out to make everything computable: encoding is a matrix product, and decoding is looking up which error pattern produces the syndrome you observed. Cyclic codes add one structural property that turns those matrix operations into shift registers.
Work through these
Linear codes, the generator matrix and the parity check matrix
A linear code is a subspace, the generator maps information into it, and the parity check matrix annihilates every valid word. Those three objects between them contain everything about the code.
NPTEL: Error Control Coding — An Introduction to Linear Block Codes (IIT Kanpur) · CourseSyndrome decoding, and the table it is really doing
Multiplying a received word by the parity check matrix gives a syndrome that depends only on the error, not on what was sent. Decoding is then a lookup from syndrome to most likely error pattern.
NPTEL: Error Control Coding — An Introduction to Linear Block Codes (IIT Kanpur) · CourseThe Hamming code, and single error correction for very little cost
A handful of parity bits placed so that the syndrome spells out the position of the error is the smallest complete example of the idea. It is still used inside memory chips for exactly that reason.
NPTEL: Error Control Coding — An Introduction to Linear Block Codes (IIT Kanpur) · CourseCyclic codes, generator polynomials, and encoding with a shift register
Requiring that any rotation of a codeword is also a codeword turns the algebra into polynomial arithmetic, which a short shift register performs directly. That is why cyclic codes dominate hardware implementations.
NPTEL: Error Control Coding — An Introduction to Linear Block Codes (IIT Kanpur) · CourseCodes over larger symbols, and why Reed–Solomon suits bursts
Working with symbols of several bits rather than single bits means a run of consecutive bit errors damages only one or two symbols. That is why this family appears in storage, in broadcast and in every optical disc.
NPTEL: Information Theory, Coding and Cryptography (IIT Delhi) · CourseShortening, puncturing and getting the rate you actually need
Standard codes come in fixed sizes, and practical systems adjust them by omitting information positions or by not transmitting some parity ones. Both operations change the distance, and by how much has to be checked.
Sign in to keep your progress.
Free resources
We haven't checked most of these for screen reader use yet.
Links last checked 4 Sept 2026.
Stuck here?
Ask a mentor. A real person answers, and they can see exactly which topic you're on. Usually within a couple of working days.
Checking your session…
Topics shown in module order.