Error-Correcting Codes

Key idea

Hamming and linear codes detect and repair errors.

In depth

A linear code is a subspace of Fⁿ. A parity-check matrix H detects errors via Hx = 0, and the syndrome Hr identifies where a received word r went wrong. Hamming codes correct a single error.

Worked example

A Hamming (7,4) code uses 3 parity bits to correct any single-bit error.

Real-world uses

• Repairing damaged data on DVDs, satellite links and computer memory

Common mistake

Expecting to correct more errors than the minimum distance allows.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Linear Algebra

• Triple Product – Scalar triple product gives parallelepiped volume.
• Homogeneous Systems – Ax = 0 always has the trivial solution.
• Cofactor Expansion – Compute determinants recursively.
• Properties of Determinants – Multiplicative; row swaps change sign.
• Geometric Meaning of Determinants – Area or volume scaling factor.
• Cramer’s Rule for Systems – Solve systems using determinants.