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Covering congruences related to modular arithmetic and error correcting codes. (English) Zbl 0535.10049
Covering system of congruences (CS) is a system of congruences \(x\equiv a_ j (mod m_ j)\), \(j=1,2,...,k\), with the property that every integer satisfies at least one of the congruences. An old problem about CS’s asks for a sufficient and necessary condition for the existence of a CS with given moduli \(m_ 1,...,m_ k\). The main result of the paper gives such a condition in terms of groups and their cosets depending only on the moduli. The result is impressive but as the author says ”it is not entirely satisfactory since the existence quantifier is used”. Nevertheless, it associates the CS’s with single-error-correcting codes, what leads to interesting reformulations of some well-known and long- standing open problems about CS’s.
Reviewer: Št.Porubský

11B99 Sequences and sets
94B05 Linear codes (general theory)
11A07 Congruences; primitive roots; residue systems