Fire Codes
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Transcript of Fire Codes
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FIRE CODES
MINIMUN POLYNOMIALSBOSE-CHAUDHURI-HOCQUENGHEM CODESREED-SOLOMON CODES
Juan Astudillo, Miguel Davila y Gustavo Delgado
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FIRE CODES
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FIRE CODES
Fire Code can correct single-burst errors in code vectors.
Definition: A Fire code is a cyclic code with a generator
polynomial of the form:
Where p(x) is an irreducible polynomial over B ofdegree m, whose roots have order r and c is not
divisible by r
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FIRE CODES
The length n of the code words in a Fire Code isthe least commom multiple of c and r, the numberof parity check bits is c+m and the number of
information bits is n-c-m.
The code is capable correcting a single burst of
length b and simultaneously detecting a burst oflength:
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FIRE CODES
Example
We have seen that is irreducible over Band that the order of its roots is 3. We can use this
polynomial to construct a generator polynomial for aFire code by multiplying it by to give:
In this case, we have c=4, m=2 and r=3. The code hascode words that are twelve bits long, with sixinformation bits and six parity check bits. It can correctbursts up to two bits long.
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FIRE CODES
Its generator matrix is:
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FIRE CODES
Example
It is easy to construct Fire codes with long codewords. Considerer:
is irreducible over B, and the order of
its roots is 127.
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FIRE CODES
We therefore have m=7, r=127 and c=8. The leastcommom of c and is 8*127=1016.
The code has code words that are 1016 bits long,with 15 parity check bits and 1001 informationbits.
It can be correct bursts up to seven bits long.
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Minimum Polynomials
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MINIMUM POLYNOMIALS
Let F be any field. For any , the polynomial (X-a)has a as zero. It is irreducible over F.
If is irreducible over F, and th
e degree ofp(X) is greater than 1, then F(X)/p(X) is field in whichthe coset of X is a zero of p(X).
There may be many polynomials that have a given
zero, some of which are irreducible and some ofwhich are not. Of these, there is one special polynomialthat has the smallest degree. To specify it, we need thefollowing definitions.
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MINIMUM POLYNOMIALS
Minimum Polynomial: Let F be a field, and let abelong either to F or an extension field of F.
If is an irreducible monic polynomial ofwhich a is a zero, and there is no polynomial of lesserdegree of which a is a zero, then p(X) is the minimum
polynomial ofa over F.
Note that if p(X) is the minimum polynomial of a over F,then p(X) will be a factor of any other polynomial ofwhich a is a zero.
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MINIMUM POLYNOMIALS
Example
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BOSE-CHAUDHURI-HOCQUENGHEM CODES
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BOSE-CHAUDHURI-HOCQUENGHEMCODES
Bose-Chaudhuri-Hocquenghem (BCH) codes arecyclic codes whose generator polynomial has beenchosen to make the distance between code words
large, and for which effective decoding procedureshave been devised.
The construction of BCH codes uses roots of unity.
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BOSE-CHAUDHURI-HOCQUENGHEMCODES
Definition: nth Root of Unity: Let F be a field. An nth root of unity is a zero of the polynomial:
1 is obviously always an n th th root of unity, but inmost cases, the roots of unity will not belong to F,but to some extension field of F. For a Galois fieldGF(p) there will be some m such that the nth
roots of unity belong to . In this case, nmust divide .
(This means that n and p cannot have any commonfactors.)
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BOSE-CHAUDHURI-HOCQUENGHEMCODES
Example:
The 3rd roots of unity of B have n=3,p=2. Since2^2-1=3, we have m=2. The roots of unity are the
three non-zero elements of Galois field. Since
the minimum polynomial of 1 is (X+1) and theminimum polynomial of the other roots of unity is
The zeros of are primitive roots ofunity in B.
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BOSE-CHAUDHURI-HOCQUENGHEMCODES
Definition: Least Common Multiple: The least common
multiple of a set of polynomials is the polynomial of
minimum degree that is divisible byall the polynomials
in the set.
Example:
In the least common multiple of X and (X+1) is
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BOSE-CHAUDHURI-HOCQUENGHEMCODES
The least common multiple of (X+1) and
since (X+1) divides The least common multiple of
can be found by finding the factors of thesepolynomials and multiplying together those that
appear in at least one of the polynomials.
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BOSE-CHAUDHURI-HOCQUENGHEMCODES
Since
their least common multiple is given by
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BOSE-CHAUDHURI-HOCQUENGHEMCODES
Definition: Bose-Chaudhuri-Hocquenghem (BCH)
Code: A Bose-Chaudhuri-Hocquenghem (BCH) codeis a cyclic code of length n whose generator
polynomial is the least common multiple of theminimal polynomials of successive powers of a
primitive n th root of unity in B.
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BOSE-CHAUDHURI-HOCQUENGHEMCODES
From the above, there is some m such that
contains a root of unity in B.
are positive integers, then
are successive powers of a. Each of these powers
willhave a minimal polynomial in B[X].
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BOSE-CHAUDHURI-HOCQUENGHEMCODES
The least common multiple of these minimal
polynomials will be the generator polynomial ofa cyclic code whose minimum distance will be no
less than .
is the designed distance of the code.
The most important BCH codes are obtained bytaking b=1.
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BOSE-CHAUDHURI-HOCQUENGHEMCODES
It can be shown that for any positive integers mand t, there is a BCH binary code of length
which corrects all combinations of t or fewer errorsand has no more than mt parity check bits. Inparticular, the code will correct bursts of length t or
less.
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BOSE-CHAUDHURI-HOCQUENGHEMCODES
Example
is irreducible over B. If welet a be the coset of X in B[X]/p(X), and take m=3,b=1 and = 3, we get BCH code whose codewords are 7 bits long.
The generator polynomial of this code is thepolynomial in B[X] of minimal degree whose roots
include
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BOSE-CHAUDHURI-HOCQUENGHEMCODES
The polynomial has this property, since
The generator matrix of the code is
The code has three information bits and four paritycheck bits.
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Reed-Solomon Codes
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Reed-Solomon Codes
Definition: Reed-Solomon Codes: A Reed-Solomon
code is a BCH code with parameters m=1 and b=1.
Reed-Solomon codes are an important subclass ofBCH codes. They are constructed in the followingmanner. Let F be a finite field, and let n be the
order of that is, The polynomial
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Reed-Solomon Codes
The generator polynomial of a code whose wordshave n digits (d-1)parity check digits andminimum distance d.
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Reed-Solomon Codes
Example
In if a denotes the coset of X,
. If we take d=4, our generator
polynomial is
If we rename the elements of
with the digits 0,1,2,3,4,5,6,7 the generatorpolynomial is
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Reed-Solomon Codes
and we have a generator matrix
For a code on {0,1,.,7}
If we express the digits in the generator matrix in
binary notation, we get the generator matrix for abinary code with code words that are 21 bits long:
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Reed-Solomon Codes
Note that if G is used to generate code words, the
operations of with theelements suitably renamed, must be used in the
computations.
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Reed-Solomon Codes
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Reed-Solomon Codes
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Reed-Solomon Codes