Even code: Difference between revisions

Revision as of 15:21, 11 April 2006

Latest revision as of 18:33, 29 April 2024

(18 intermediate revisions by 15 users not shown)

Line 1:

A{{Short [[binary code]] is called an '''even code''' ifdescription|If the [[Hamming weight]] of all its codewords is even. Furthermore,of a binary code is called [[doubly-even]] if the Hamming weight of all its's codewords is divisible by 4. An even code which is not doubly-even is said to be strictly even.}}

{{One source|date=October 2023}}

A [[linear code|binary code]] is called an '''even code''' if the [[Hamming weight]] of each of its codewords is even. An even code should have a generator polynomial that include (1+''x'') minimal polynomial as a product. Furthermore, a binary code is called '''doubly even''' if the Hamming weight of all its codewords is [[divisible by 4]]. An even code which is not doubly even is said to be strictly even.

Examples of doubly- even codes are the extended binary [[Hamming code]] of [[block length]] 8 and the [[extended binary [[Binary Golay code|Golay code]] of block length 24. These two codes are, in addition, [[self-dual code|self-dual]].

{{planetmathPlanetMath attribution|id=7047|title=even code}}

[[Category:Coding theory]]

{{math-stub}}

[[Category:Parity (mathematics)]]

{{mathcrypto-stub}}