Difference between revisions of "Linear code"
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A [[code]] of fixed length $n$ over a finite field $F$ which forms a subspace of the vector space $F^n$. | A [[code]] of fixed length $n$ over a finite field $F$ which forms a subspace of the vector space $F^n$. | ||
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| + | A linear code of rank $r$ may be represented by a ''generator matrix'', an $(r \times n)$ matrix whose rows form a set of linearly independent code words. A generator matrix can be put in the form $G = (I_r | G_1)$ by row operations, showing that a linear code is necessarily a [[systematic code]]. The components of the code word corresponding to the columns of $G_1$ may be referred to as check digits. | ||
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| + | A linear code $C$ has a '''dual code''' $C^\perp$ consisting of all vectors in $F^n$ orthogonal to every element of $C$ with respect to the bilinear form $(x,y) = \sum_{i=1}^n x_i y_i$. This is a linear code of rank $(n-r)$, and a basis for $C^\perp$ is a set of ''parity check'' vectors: a generator matrix for $C^\perp$ is a '''parity check matrix'''. | ||
See [[Error-correcting code]]. | See [[Error-correcting code]]. | ||
====References==== | ====References==== | ||
| − | * Goldie, Charles M.; Pinch, Richard G.E. ''Communication theory'', London Mathematical Society Student Texts. '''20''' Cambridge University Press (1991) | + | * Goldie, Charles M.; Pinch, Richard G.E. ''Communication theory'', London Mathematical Society Student Texts. '''20''' Cambridge University Press (1991) {{ISBN|0-521-40456-8}} {{ZBL|0746.94001}} |
| + | * van Lint, J.H., "Introduction to coding theory" (2nd ed.) Graduate Texts in Mathematics '''86''' Springer (1992) {{ISBN|3-540-54894-7}} {{ZBL|0747.94018}} | ||
Latest revision as of 20:28, 15 November 2023
A code of fixed length $n$ over a finite field $F$ which forms a subspace of the vector space $F^n$.
A linear code of rank $r$ may be represented by a generator matrix, an $(r \times n)$ matrix whose rows form a set of linearly independent code words. A generator matrix can be put in the form $G = (I_r | G_1)$ by row operations, showing that a linear code is necessarily a systematic code. The components of the code word corresponding to the columns of $G_1$ may be referred to as check digits.
A linear code $C$ has a dual code $C^\perp$ consisting of all vectors in $F^n$ orthogonal to every element of $C$ with respect to the bilinear form $(x,y) = \sum_{i=1}^n x_i y_i$. This is a linear code of rank $(n-r)$, and a basis for $C^\perp$ is a set of parity check vectors: a generator matrix for $C^\perp$ is a parity check matrix.
References
- Goldie, Charles M.; Pinch, Richard G.E. Communication theory, London Mathematical Society Student Texts. 20 Cambridge University Press (1991) ISBN 0-521-40456-8 Zbl 0746.94001
- van Lint, J.H., "Introduction to coding theory" (2nd ed.) Graduate Texts in Mathematics 86 Springer (1992) ISBN 3-540-54894-7 Zbl 0747.94018
Linear code. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_code&oldid=39142