When you send a sequence
of bits from point A to point B, you want to know that it will arrive
without error. A common form of insurance is the parity bit, attached
to 7-bit ASCII characters to put them into 8-bit format. The parity bit
is chosen so as to make the total number of one-bits (versus zero-bits)
either always even (even parity or always odd (odd parity . Any
Unfortunately, in real
situations, a single noise event is likely to disrupt more than one
bit. Since the parity bit has two possible values (0 and 1), it gives,
on average, only a 50% chance of detecting an erroneous character with
more than one wrong bit. That probability, 50%, is not nearly good
enough for most applications. Most communications protocols use a
multibit generalization of the parity bit called a cyclic redundancy
check or CRC. In typical applications the CRC is 16 bits long (two
bytes or two characters), so that the chance of a random error going
undetected is 1 in
▪ To show the intermediate step in encoding and decoding, and allow the use of different encoding polynomials. ▪ To find out whether the used one is a generator polynomial or not.
Now let us briefly discuss the theory of CRCs. After that, we will give implementations of various (related) CRCs that are used by the official. The mathematics underlying CRCs is polynomials over the integers modulo 2. Any binary message can be thought of as a polynomial with coefficients 0 and 1.
For example, the message
1100001111 is the polynomial
g(x)
is the generator polynomial for a linear cyclic code of length n if and
only if g(x) divides 1 + x
What this is saying is
that to find generator polynomial g(x) If n is the length of the code the length of the code, k = n - deg(g(x)) where deg(g(x)) denotes the degree of g(x).
To Find a generator
polynomial and for a code of length n = 15 which will encode messages of
length k = 7. Then encode the word [0110110]. If we are to find a
generator for a code of length 15 to encode messages of length 7.
The polynomial x
x
so
we can choose g(x) = (1 + x + x
(x + x
x + x Thus the word [0110110] is encoded to the codeword [011010111100010].
In this project, I use the Java language for implementing the codes. So you can find the results in Here!.
Multiplication can be implemented by shift registers and exclusive-or gates, so in my project, I also use this scheme for encoding. (See ccStatus.java in Sources section)
Division or decoding can be done by multiplication in the feedback loop, of course this benefit of this scheme is re-usability of the encoding scheme, so that we can reduce many of duplicating codes. If no bit errors exist we will receive E(X) and calculate D(X) by D(X)=E(X)/G(X), reminder is zero.
Example: E(X)=D(X)G(X)=D(X)(
= D(X)+D(X)(
D(X)=-E(X)+
D(X)(
If and only if g(x)
divides 1 + x
So, I temporarily set
the Input bit(k) and Code bit(n) in decoding mode, and then check
whether g(x) can divide 1+ x |