Namespaces
Variants
Actions

Difference between revisions of "User:Richard Pinch/sandbox-7"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Start article: Weight enumerator)
 
Line 1: Line 1:
=Weight enumerator=
+
=Distance enumerator=
  
The distribution of [[Hamming weight]]s of a [[code]] expressed as a polynomial.  Let $C$ be a code of length $n$ over an alphabet $F$ and let $A_k$ be the number of of words of $C$ of weight $k$.  The weight enumerator
+
The distribution of [[Hamming distance]]s between elements of a [[code]], expressed as a polynomial.  Let $C$ be a code of length $n$ over an alphabet $F$ and let $A_k$ be the number of pairs $x,y$ of words of $C$ of at Hamming distance $d(x,y) = k$.  The weight enumerator
 +
$$
 +
W_C(z) = \sum_{k=0}^n A_k z^k = \sum_{x,y \in C} z^{d(x,y)} \ .
 +
$$
 +
It is also common to express the weight enumerator as a homogeneous binary form
 +
$$
 +
W_C(x,y) = \sum_{k=0}^n A_k x^k y^{n-k} \ .
 +
$$
 +
 
 +
We have $W_C(0) = |C|$ and $W_C(1) = |C|^2$, where $|C|$ is the number of words in $C$.
 +
 
 +
The '''weight enumerator''' similarly expresses the distribution of [[Hamming weights]]s of elements of a [[code]], expressed as a polynomial.  Let $C$ be a code of length $n$ over an alphabet $F$ and let $A_k$ be the number of of words of $C$ of weight $k$.  The weight enumerator
 
$$
 
$$
 
W_C(z) = \sum_{k=0}^n A_k z^k = \sum_{x \in C} z^{w(x)}
 
W_C(z) = \sum_{k=0}^n A_k z^k = \sum_{x \in C} z^{w(x)}
Line 10: Line 21:
 
$$
 
$$
  
We have $W_C(0) = 0$ or $1$, depending on whether the zero word is in $C$ or not, and $W_C(1) = |C|$, the number of words in $C$.
+
We have $W_C(0) = 1$ or $0$, depending on whether the zero word is in $C$ or not, and $W_C(1) = |C|$, the number of words in $C$.
  
 
The [[MacWilliams identities]] relate the weight enumerator of a linear code over a finite field $\mathbf{F}_q$ to the enumerator of the dual code $C^\perp$:
 
The [[MacWilliams identities]] relate the weight enumerator of a linear code over a finite field $\mathbf{F}_q$ to the enumerator of the dual code $C^\perp$:

Revision as of 20:32, 17 September 2016

Distance enumerator

The distribution of Hamming distances between elements of a code, expressed as a polynomial. Let $C$ be a code of length $n$ over an alphabet $F$ and let $A_k$ be the number of pairs $x,y$ of words of $C$ of at Hamming distance $d(x,y) = k$. The weight enumerator $$ W_C(z) = \sum_{k=0}^n A_k z^k = \sum_{x,y \in C} z^{d(x,y)} \ . $$ It is also common to express the weight enumerator as a homogeneous binary form $$ W_C(x,y) = \sum_{k=0}^n A_k x^k y^{n-k} \ . $$

We have $W_C(0) = |C|$ and $W_C(1) = |C|^2$, where $|C|$ is the number of words in $C$.

The weight enumerator similarly expresses the distribution of Hamming weightss of elements of a code, expressed as a polynomial. Let $C$ be a code of length $n$ over an alphabet $F$ and let $A_k$ be the number of of words of $C$ of weight $k$. The weight enumerator $$ W_C(z) = \sum_{k=0}^n A_k z^k = \sum_{x \in C} z^{w(x)} $$ where $w(x)$ is the weight of the word $x$. It is also common to express the weight enumerator as a homogeneous binary form $$ W_C(x,y) = \sum_{k=0}^n A_k x^k y^{n-k} \ . $$

We have $W_C(0) = 1$ or $0$, depending on whether the zero word is in $C$ or not, and $W_C(1) = |C|$, the number of words in $C$.

The MacWilliams identities relate the weight enumerator of a linear code over a finite field $\mathbf{F}_q$ to the enumerator of the dual code $C^\perp$: $$ W_{C^\perp}(x,y) = \frac{1}{|C|} W_C(x + (q-1)y, x-y) \ . $$

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
How to Cite This Entry:
Richard Pinch/sandbox-7. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-7&oldid=39147