PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (194)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
perfect code (Definition)

Let $ C$ be a linear $ (n,k,d)$-code over $ \mathbb{F}_q$.

The packing radius of $ C$ is defined to be the value

$\displaystyle \rho(C)=\frac{d-1}{2}.$    

The covering radius of $ C$ is

$\displaystyle r(C)=\max_x\min_c \delta(x,c)$    

with $ x\in \mathbb{F}_q^n$ and $ c\in C$, and where $ \delta$ denotes the Hamming distance on $ \mathbb{F}_q^n$.

The code $ C$ is said to be perfect if $ r(C)=\rho(C)$.

The list of classes of linear perfect codes is very short, including only trivial codes, Hamming codes (i.e. $ \rho=1$), and the binary and ternary Golay codes.



"perfect code" is owned by mathcam.
(view preamble)

View style:

Also defines:  packing radius, covering radius
Log in to rate this entry.
(view current ratings)

Cross-references: binary, codes, perfect, Hamming distance

This is version 2 of perfect code, born on 2004-06-04, modified 2004-06-08.
Object id is 5892, canonical name is PerfectCode.
Accessed 3193 times total.

Classification:
AMS MSC: 11T71 (Number theory :: Finite fields and commutative rings :: Algebraic coding theory; cryptography)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)