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Zbl 1181.16024
Kelarev, A.V.; Yearwood, J.L.; Vamplew, P.W.
A polynomial ring construction for the classification of data.
(English)
[J] Bull. Aust. Math. Soc. 79, No. 2, 213-225 (2009). ISSN 0004-9727

Let $\bbfF[X]/I_P$ be the factor algebra of the polynomial algebra in $m$ variables over the finite field $\bbfF$ of characteristic $p$ modulo the ideal $I_P$ generated by the differences of monomials $p-q$, $(p,q)\in P\subset [X]\times [X]$. Assuming that the ideal $I_P$ is of codimension $N$, the authors identify $\bbfF[X]/I_P$ with $\bbfF^N$. Let $C=C(U)$ be the ideal of $\bbfF^N$ generated by the set $U\subset\bbfF^N$. The authors call $U$ a visible set of generators of $C$ if the minimal Hamming weight of $C$ is equal to that of $U$.\par In 1988 the reviewer and Lakatos introduced a class of ideals of the factor algebra $\bbfF[X]/(x_i^p-1$, $i=1,\dots,m)$ with visible sets of generators. The class includes several important error correcting codes realized as ideals of modular group algebras.\par In the paper under review the authors extend essentially the class of the reviewer and Lakatos to ideals of the algebra $\bbfF[X]/I_P$ when the commutative semigroup $[X]/(p=q$, $(p,q)\in P)$ is a subsemigroup of a direct product of a semilattice, an elementary Abelian 2-group and an elementary Abelian $p$-group. The authors give examples which show that their results cannot be extended to larger classes of ring constructions and cannot be simplified or generalized. The main results are considered from the point of view of applications to design multiple classifiers and to use them to correct errors of the individual classifiers which constitute the multiple ones.
[Vesselin Drensky (Sofia)]
MSC 2000:
*16S36 Ordinary and skew polynomial rings and semigroup rings
20M25 Semigroup rings, multiplicative semigroups of rings
16S34 Group rings (assoc. rings)
16D25 2-sided ideals (assoc. rings and algebras)
94B60 Other types of codes
68Q45 Formal languages
68T10 Pattern recognition

Keywords: ring constructions; group rings; classification of data; error-correcting capability; combined multiple classifiers; visible sets of generators

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