*(English)*Zbl 1181.16024

Let $\mathbb{F}\left[X\right]/{I}_{P}$ be the factor algebra of the polynomial algebra in $m$ variables over the finite field $\mathbb{F}$ of characteristic $p$ modulo the ideal ${I}_{P}$ generated by the differences of monomials $p-q$, $(p,q)\in P\subset \left[X\right]\times \left[X\right]$. Assuming that the ideal ${I}_{P}$ is of codimension $N$, the authors identify $\mathbb{F}\left[X\right]/{I}_{P}$ with ${\mathbb{F}}^{N}$. Let $C=C\left(U\right)$ be the ideal of ${\mathbb{F}}^{N}$ generated by the set $U\subset {\mathbb{F}}^{N}$. The authors call $U$ a visible set of generators of $C$ if the minimal Hamming weight of $C$ is equal to that of $U$.

In 1988 the reviewer and Lakatos introduced a class of ideals of the factor algebra $\mathbb{F}\left[X\right]/({x}_{i}^{p}-1$, $i=1,\cdots ,m)$ with visible sets of generators. The class includes several important error correcting codes realized as ideals of modular group algebras.

In the paper under review the authors extend essentially the class of the reviewer and Lakatos to ideals of the algebra $\mathbb{F}\left[X\right]/{I}_{P}$ when the commutative semigroup $\left[X\right]/(p=q$, $(p,q)\in P)$ 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.

##### MSC:

16S36 | Ordinary and skew polynomial rings and semigroup rings |

20M25 | Semigroup rings, multiplicative semigroups of rings |

16S34 | Group rings (associative rings), Laurent polynomial rings |

16D25 | 2-sided ideals (associative rings and algebras) |

94B60 | Other types of codes |

68Q45 | Formal languages and automata |

68T10 | Pattern recognition, speech recognition |