Simplifications to A New Approach to the Covering Radius...”

We simplify the proofs of four results in [3], restating two of them for greater clarity. The main purpose of this note is to give a brief transparent proof of Theorem 7 of [3], the main upper bound of that paper. The secondary purpose is to give a more direct statement and proof of the integer programming determination of covering radius of [3]. Theorem 7 of [3] follows from a simple result in [2], which we state with the notation (for the linear code A)

New upper bounds for the football pool problem for 11 and 12 matches

AbstractWe consider the problem of minimizing the number of words in a code with the property that all words in the space F3n are within Hamming distance 1 from some codeword. This problem is called the football pool problem, since the words in such a code can be used in a football pool to guarantee that at least one forecast has at least n − 1 correct results. In this note we show that for 11 and 12 matches, there are 9477 and 27702 words, respectively, having the aforementioned property. Simulated annealing has played an important role in the search for these words

On the normality of multiple covering codes

AbstractA binary code C of length n is called a μ-fold r-covering if every binary word of length n is within Hamming distance r of at least μ codewords of C. The normality and the amalgamated direct sum (ADS) construction of 1-fold coverings have been extensively studied. In this paper we generalize the concepts of subnormality and normality to μ-fold coverings and discuss how the ADS construction can be applied to them. In particular, we show that for r = 1, 2 all binary linear μ-fold r-coverings of length at least 2r + 1 and μ-fold normal

Fast generation and covering radius of Reed-Muller Codes

Reed-Muller codes are known to be some of the oldest, simplest and most elegant error correcting codes. Reed-Muller codes were invented in 1954 by D. E. Muller and I. S. Reed, and were an important extension of the Hamming and Golay codes because they gave more flexibility in the size of the codeword and the number of errors that could be correct. The covering radius of these codes, as well as the fast construction of covering codes, is the main subject of this thesis. The covering radius problem is important because of the problem of constructing codes having a specified length and dimension. Codes with a reasonably small covering radius are highly desired in digital communication environments. In addition, a new algorithm is presented that allows the use of a compact way to represent Reed-Muller codes. Using this algorithm, a new method for fast, less complex, and memory efficient generation of 1st and 2nd order Reed - Muller codes and their hardware implementation is possible. It is also allows the fast construction of a new subcode class of 2nd order Reed-Muller codes with good properties. Finally, by reversing this algorithm, we introduce a code compression method, and at the same time a fast, efficient, and promising error-correction process.http://archive.org/details/fastgenerationnd109454471Hellenic Army author

Covering Radius 1985-1994

We survey important developments in the theory of covering radius during the period 1985-1994. We present lower bounds, constructions and upper bounds, the linear and nonlinear cases, density and asymptotic results, normality, specific classes of codes, covering radius and dual distance, tables, and open problems