A
binary code
A binary code represents text, computer processor instructions, or any other data using a two-symbol system. The two-symbol system used is often "0" and "1" from the binary number system. The binary code assigns a pattern of binary digits, also ...
is called an even code if the
Hamming weight
The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string o ...
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
In computer science and telecommunication, Hamming codes are a family of linear error-correcting codes. Hamming codes can detect one-bit and two-bit errors, or correct one-bit errors without detection of uncorrected errors. By contrast, the sim ...
of block length 8 and the extended binary
Golay code of block length 24. These two codes are, in addition,
self-dual
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involutio ...
.
{{crypto-stub
Coding theory
Parity (mathematics)