The Weight Enumerator of GRM codes

: The distribution of weight enumerator gives the new idea to contract the information about the code for finding the error probability of a code. In this paper, we have extended the result of the weight enumerator of GRM codes [3]. The weight enumerator of GRM codes for order )3 ( - m , m 0 ,3 ³ ³ r has been developed and analyzed.

And most of studies must be limited to the detection and correction of random error in coding theory. Fire (1959) design 'burst errors' codes which later become useful for correction of BCH and many burst errors which is already known. In 1960, Abramson created binary codes that could correct all single and double adjacent-errors which later become the cornerstone of an important class in coding theory called 'burst error correcting codes' and the occurrence of errors was found more in the pedestal position rather than randomly in practical channels, so this code was used extensively.
In this paper we worked and focused on Generalized Reed Muller (GRM) codes developed from RM codes by Dass and Muttoo by extending/ shortening the RM codes. RM codes were developed by Muller whereas Reed provided the first effective translation method of these codes. Reed Muller codes are best, oldest and well ancestry of error correcting codes. It forms infinite family of codes, and its important merit is that larger Reed Muller codes can be made from smaller code. The main advantage of Reed Muller code is to encode the message and decode the broadcast received. For this, we use the generator matrices for encoding and decoding using one form of process known as majority logic. In this correction of information transmission, the code has value both on earth and from space. Reed Muller code efficient of correcting 7 errors out of 32 bits communicated, including now of 6 data bits and 26 check bits over 16,000 bits per second were import back to earth. Special cases of Reed Muller codes include the Walsh-Hadamard code [14] and Reed-Soloman codes [8] The weight distribution of second order q-ary Reed Muller codes has been developed by Sloane and Berlekamp [11]. Further a precise account of weight distribution of second order q-ary Reed Muller codes provided by Shuxing Li [18] with some corrections and also in addition, second order q-ary projective Reed Muller codes and the weight distributions of these codes are homogeneous.
In [6], by taking the advantage of fast Fourier techniques, for a given binary non-linear code a deterministic algorithm is provided to compute distance distribution, its weight, minimum distance and minimum weight. The performance of above algorithm was found to be similar as that of best known algorithm for average case, which is specifically efficient for codes with low information rate. Also in [15], for q-ary simplex code and q-ary first order reed Muller code the extended and generalized weight enumerator has already been studied and for calculation purpose all the codes were used as they corresponds to a projective system containing all the points in a finite projective or affine space. So, weight enumeration has been from the geometric method.
In [12], computed the weight enumerator of affine Reed Muller codes and some projective of order 3 over q F which give the answer of enumerative questions about plane cubic curves. Also, it is found that how traces of Hecke operators performing on spaces of cusp forms for ) ( 2 Z SL . In this paper, we used the most fundamental outcome about weight distribution that are MacWilliams equation, a set of linear relation between the weight distribution of code ) (c and dual of code ) (ĉ . These equations suggested that the weight distribution of ) (ĉ is uniquely determined by weight distribution of c . This paper has four Sections. Section 1, contains important definitions used in the paper. In Section 2, theorem on weight enumerator of GRM codes of order (m-3) is stated and proved. In the last Section, conclusion is given.

Section 2
The weight distribution is a fundamental parameter of Reed Muller codes. Their essential importance as mathematical objects, are mostly used in probability theory to bring in with different ways of decoding. In this paper, we have obtained the weight enumerator of the

= = m A A
Therefore, we get the formula for Now, using the result, known as Macwilliam identity [7], we get the weight enumerator of , where the rows of the matrix X, representing the product of some s vector of representing the linear combination of the rows of X .The code Let, the weight enumerator of a coset ) , In 1978, Assumus and Mattson [1] developed a result on the weight enumerator of a coset of a linear code. Using the result, we will get ú û Where, the number of code words of weight i in From the expression of ) ( X G it follows that , , , For .
From this, it follows that From equation (1) and (3) and (4)

Section 4 Conclusion:
The work has already been done up to order (m-2) and in this paper, we have extended this work for the weight enumerator of Generalized Reed Muller codes of order ) 3 (m .