Bch Rs Codes

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Ass.Prof.Dr.Thamer Information Theory 4th Class in Communications

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BCH Codes

The Bose, Chaudhuri, and Hocquenghem (BCH) codes form

a large class of powerful random error-correcting cyclic codes.This class of codes is a remarkable generalization of the Hammingcodes for multiple-error correction. Binary BCH codes werediscovered by Hocquenghem in 1959 and independently by Boseand Chaudhuri in 1960 .

For any positive integers m(m 3) and t (t< 2m-1), thereexists a binary BCH code with the following parameters:

Clearly, this code is capable of correcting any combination oft or fewer errors in a block of n = 2m l digits. We call this

code a t-error-correcting BCH code.The generator polynomial of this code is specified in terms

of its roots from the Galois field GF(2m). Let be a primitive

element in GF(2m). The generator polynomial g(X) of the t-error-correcting BCH code of length 2m l is the lowest-degreepolynomialover GF(2) which has

.(1)

As its roots [i.e., g(i) = 0 for 1 i 2t . It follows that

g(X) has, 2, . . . , 2t and their conjugates as all its roots.


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Let i(X) be the minimal polynomial ofi. Then g(X) must

be the least common multiple (LCM) of1(X) , 2(X) , ..2t (X), that is,

(2)The generator polynomial g(X) of the binary t-error-

correcting BCH code of length 2m 1 given by (2) can bereduced to

(3)

Since the degree of each minimal polynomial is m or less,the degree ofg (X) is at most mt . That is, the number of parity-check digits, n k, of the code is at most equal to mt .

There is no simple formula for enumerating n k, but ift

is small, n

k is exactly equal to mt. The parameters for allbinary BCH codes of length 2m 1 with m 10 , are given inTable 1. The BCH codes defined above are usually calledpr imit ive(or narrow-sense) BCH codes.


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Example 1: Let a be a primitive element of the Galois field

GF(24) given by Table 2 , such that 1 + + 4 = 0. From

Table 3 , we find that the minimal polynomials of, 3, and 5

are


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It follows from (3) that the double-error-correcting BCH code oflength n = 24 1=15 is generated by

Thus, the code is a (15, 7) cyclic code with dm i n 5. Since the

generator polynomial is code polynomial of weight 5, theminimum distance of this code is exactly 5.

EXAMPLE 2: The triple-error-correcting BCH code of length 15is generated by

This triple-error-correcting BCH code is a (15, 5) cyclic code withdm in 7. Since the weight of the generator polynomial is 7, theminimum distance of this code is exactly 7.

Using the primitive polynomial p(X) = 1 + X + X6, wemay construct the Galois field GF(26) as shown in Table 4. Theminimal polynomials of the elements in GF(26) are listed in Table


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5. Using (3), we find the generator polynomials of all the BCHcodes of length 63 as shown in Table 6.


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H matrix (i.e. the parity-check matrix) of the BCH code isgiven as

(4)Note that the entries ofH are elements in GF(2m). Each elementin GF(2m) can be represented by a w-tuple over GF(2). If eachentry of H is replaced by its corresponding m-tuple over GF(2)arranged in column form, we obtain a binary parity-check matrixfor the code.

Example 3: Consider the double-error-correcting BCH code of

length n = 2" 1 =15. From Example 1 we know that this is a(15, 7) code. Let be a primitive element in GF(24). Then the

parity-check matrix of this code isUsing (4) we get that:


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Using Table 2, the fact that 15 = 1, and representing

each entry ofH by its corresponding 4-tuple, we obtainthe following binary parity-check matrix for the code:

The t-error-correcting BCH code denned above indeed hasminimum distance at least 2t + 1. The parameter 2t + 1 isusually called the designed distance of the /-error-correcting

BCH code. The true minimum distance of a BCH code may ormay not be equal to its designed distance. There are many caseswhere the true minimum distance of a BCH code is equal to itsdesigned distance. However, there are also cases where the trueminimum distance is greater than the designed distance.


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REED-SOLOMON CODES

In addition to the binary codes, there are non binary codes. In

fact, ifp is a prime number and q is any power ofp, there arecodes with symbols from the Galois field GF(q). These codesare called q-ary codes .The special subclass ofq-ary BCH codes for which s = 1 is themost important subclass of q-ary BCH codes. The codes of thissubclass are usually called the Reed-Solomoncodes in honor oftheir discoverers . A t-error-correcting Reed-Solomon code withsymbols from GF(q) has the following parameters:

We consider Reed-Solomon codes with code symbols from the

Galois field GF(2m) (i.e., q = 2m). Let be a primitive element in

GF(2m). The generator polynomial of a primitive t-error-correcting Reed-Solomon code of length 2m 1 is

(5)


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The code generated by g(X) is an (n, n 2t ) cyclic code whichconsists of those polynomials of degree n 1 or less withcoefficients from GF(2m) that are multiples ofg(X). Encoding ofthis code is similar to the binary case.

EXAMPLE: Consider a triple-error-correcting Reed-Solomon codewith symbols from GF(24). The generator polynomial of this codeis

H.W: Find the generator polynomial g(X) for a double-errorcorrecting Reed- Salmon Code (RS) with GF(8) which aregenerated by X3 + X2 + 1?

Bch Rs Codes

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