Bounds on the Parameters of Codes

Abstract

In this chapter, we establish conditions on the parameters of codes. In the first part of the chapter, we present bounds that relate between the length n , size M , minimum distance d , and the alphabet size q of a code. Two of these bounds—the Singleton bound and the sphere-packing bound—imply necessary conditions on the values of n, M, d , and q , so that a code with the respective parameters indeed exists. We also exhibit families of codes that attain each of these bounds. The third bound which we present—the Gilbert–Varshamov bound—is an existence result: it states that there exists a linear [ n, k, d ] code over GF( q ) whenever n, k, d , and q satisfy a certain inequality. Additional bounds are included in the problems at the end of this chapter. We end this part of the chapter by introducing another example of necessary conditions on codes—now in the form of MacWilliams' identities, which relate the distribution of the Hamming weights of the codewords in a linear code with the respective distribution in the dual code. The second part of this chapter deals with asymptotic bounds, which relate the rate of a code to its relative minimum distance δ = d/n and its alphabet size, as the code length n tends to infinity. In the third part of the chapter, we shift from the combinatorial setting of ( n, M, d ) codes to the probabilistic framework of the memoryless q -ary symmetric channel.