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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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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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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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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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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Since

their least common multiple is given by

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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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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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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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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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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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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

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Fire Codes