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Codes from difference sets. (English) Zbl 1367.94005

Hackensack, NJ: World Scientific (ISBN 978-981-4619-35-6/pbk; 978-981-4619-37-0/ebook). xi, 341 p. (2014).
A difference set in a group \(G\) is a \(k\)-subset \(D\) of \(G\) such that every non-zero element \(g\) in \(G\) has exactly \(\lambda\) representations \(d-d'=g\) with \(d,d'\in D\). There are several books on difference sets, see [E. H. Moore and H. S. Pollatsek, Difference sets. Connecting algebra, combinatorics, and geometry. Providence, RI: American Mathematical Society (2013; Zbl 1367.05001)] for a more recent one. A very good reference is also [T. Beth et al., Design theory. Vol. I. 2nd ed. Cambridge: Cambridge University Press (1999; Zbl 0945.05004); Design theory. Vol. I. 2nd ed. Cambridge: Cambridge University Press (1999; Zbl 0945.05004)]. Similarly, there are many books on codes, the classical one is [F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes. Parts I, II. (3rd repr.). Amsterdam etc.: North- Holland (Elsevier) (1985; Zbl 0657.94010)].
Difference sets can be used to construct designs (see [Beth et al., loc. cit.]), and the most obvious way to construct a code from a difference set is to consider the row space generated by an incidence matrix of this design. If \(G\) is cyclic, these codes are cyclic. A more subtle way is to constrcut codebooks using characters of the group \(G\), applied to the elements in \(D\).
The book is not a text book on difference sets or codes, but it is an encyclopedia on codes constructed from difference sets and their generalizations. All the known classes of difference sets which are of interest in coding theory are considered. In this way, the book is also a very valuable source for people working on difference sets since the most important constructions of difference sets are described. The book also contains short introductions to the necessary algebraic structures used in difference set theory (finite fields, number theory, characters). There is also a brief introduction to designs and (linear, cyclic) codes.
The book also contains extensive tables about the best known binary and ternary cyclic codes.
Several highly interesting research problems are scattered over the text, and I am sure that these will stimulate further research.
A difference set in a group \(G\) is a \(k\)-subset \(D\) of \(G\) such that every non-zero element \(g\) in \(G\) has exactly \(\lambda\) representations \(d-d'=g\) with \(d,d'\in D\). There are several books on difference sets, see [E. H. Moore and H. S. Pollatsek, Difference sets. Connecting algebra, combinatorics, and geometry. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1367.05001)] for a more recent one. A very good reference is also [Beth et al., loc. cit.]. Similarly, there are many books on codes, the classical one is [F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes. Parts I, II. (3rd repr.). Amsterdam etc.: North- Holland (Elsevier) (1985; Zbl 0657.94010)].
Difference sets can be used to construct designs (see [Beth et al., loc. cit.]), and the most obvious way to construct a code from a difference set is to consider the row space generated by an incidence matrix of this design. If \(G\) is cyclic, these codes are cyclic. A more subtle way is to construct codebooks using characters of the group \(G\), applied to the elements in \(D\).
The book is not a text book on difference sets or codes, but it is an encyclopedia on codes constructed from difference sets and their generalizations. All the known classes of difference sets which are of interest in coding theory are considered. In this way, the book is also a very valuable source for people working on difference sets since the most important constructions of difference sets are described. The book contains short introductions to the necessary algebraic structures used in difference set theory (finite fields, number theory, characters), too. There is also a brief introduction to designs and (linear, cyclic) codes.
The book also contains extensive tables about the best known binary and ternary cyclic codes.
Several highly interesting research problems are scattered over the text, and I am sure that these will stimulate further research.

MSC:

94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
94B05 Linear codes (general theory)
94B15 Cyclic codes
94C30 Applications of design theory to circuits and networks
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05B25 Combinatorial aspects of finite geometries
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