Section 28
Linear codes and error correction
> Algebraic application of vector spaces and finite fields. This section introduces the mathematical core of linear codes without turning the course into a full Coding Theory subject.
A linear code of length over is a vector subspace
Equivalent form
A set of codewords closed under addition and scalar multiplication over the finite field.
is called a linear code .
Equivalent form
Converts a message of symbols into a codeword of symbols.
A message is encoded by
A parity-check matrix satisfies, for every codeword
Equivalent form
Moreover, with compatible conventions for and
Defines linear constraints that every valid codeword must satisfy.
If is a valid codeword
Equivalent form
Summarizes which parity constraints a received word fails.
If , then
Equivalent form
Counts how many positions differ between two words.
The Hamming weight is
Equivalent form
Equivalent form
For a linear code
The separation between codewords determines the theoretical detection and correction capability.
A code with minimum distance can detect up to
errors and correct up to
errors in the classical symbol-error model.
Fraction of symbols in the encoded word that represent independent information.