Notes on Nonzero--Element Distribution of Maximum Distance Separable Codes.

Error-correcting codes are introduced widely as a technique to improve the reliability of the digital communication system. Let q be a prime power. The error-correcting codes over CF (q) can be used to correct or detect errors in a (q - 1)-adic numbers system, when only those codewords not having a specific element of CF (q) as the components are utilized. However, the problem concerning the number of code words that do not have a specific nonzero element as the component, has not been answered. Neither has there been a study on this problem. This paper introduces a new concept of the nonzero-element distribution for the codes over GF (q). This distribution indicates how a specific nonzero element of GF (q) distributes among the components of the codewords. Then the maximum distance separable codes over GF (q) is considered, and their nonzero-element distributions are discussed. As a result, a recurrence formula is derived which is useful in determining the nonzero-element distribution of the maximum distance separable codes. For several classes of Reed-Solomon codes, which are the very important members of the maximum distance separable codes, the formulae are derived which specify the nonzero-element distributions. [ABSTRACT FROM AUTHOR]