Upper bounds on the length function for covering codes with covering radius $ R $ and codimension $ tR+1 $

<p style='text-indent:20px;'>The length function <inline-formula><tex-math id="M3">\begin{document}$ \ell_q(r,R) $\end{document}</tex-math></inline-formula> is the smallest length of a <inline-formula><tex-math id="M4">\begin{document}$ q $\end{document}</tex-math></inline-formula>-ary linear code with codimension (redundancy) <inline-formula><tex-math id="M5">\begin{document}$ r $\end{document}</tex-math></inline-formula> and covering radius <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula>. In this work, new upper bounds on <inline-formula><tex-math id="M7">\begin{document}$ \ell_q(tR+1,R) $\end{document}</tex-math></inline-formula> are obtained in the following forms:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{split} &amp;(a)\; \ell_q(r,R)\le cq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &amp;\phantom{(a)\; } q\;{\rm{ is \;an\; arbitrary \;prime\; power}},\; c{\rm{ \;is\; independent \;of\; }}q. \end{split} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \begin{split} &amp;(b)\; \ell_q(r,R)&lt; 3.43Rq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &amp;\phantom{(b)\; } q\;{\rm{ is \;an\; arbitrary\; prime \;power}},\; q\;{\rm{ is \;large\; enough}}. \end{split} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>In the literature, for <inline-formula><tex-math id="M8">\begin{document}$ q = (q')^R $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M9">\begin{document}$ q' $\end{document}</tex-math></inline-formula> a prime power, smaller upper bounds are known; however, when <inline-formula><tex-math id="M10">\begin{document}$ q $\end{document}</tex-math></inline-formula> is an arbitrary prime power, the bounds of this paper are better than the known ones.</p><p style='text-indent:20px;'>For <inline-formula><tex-math id="M11">\begin{document}$ t = 1 $\end{document}</tex-math></inline-formula>, we use a one-to-one correspondence between <inline-formula><tex-math id="M12">\begin{document}$ [n,n-(R+1)]_qR $\end{document}</tex-math></inline-formula> codes and <inline-formula><tex-math id="M13">\begin{document}$ (R-1) $\end{document}</tex-math></inline-formula>-saturating <inline-formula><tex-math id="M14">\begin{document}$ n $\end{document}</tex-math></inline-formula>-sets in the projective space <inline-formula><tex-math id="M15">\begin{document}$ \mathrm{PG}(R,q) $\end{document}</tex-math></inline-formula>. A new construction of such saturating sets providing sets of small size is proposed. Then the <inline-formula><tex-math id="M16">\begin{document}$ [n,n-(R+1)]_qR $\end{document}</tex-math></inline-formula> codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called "<inline-formula><tex-math id="M17">\begin{document}$ q^m $\end{document}</tex-math></inline-formula>-concatenating constructions") for covering codes to obtain infinite families of codes with growing codimension <inline-formula><tex-math id="M18">\begin{document}$ r = tR+1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M19">\begin{document}$ t\ge1 $\end{document}</tex-math></inline-formula>.</p>

The sub-class sizes of some elements being square free

Let [Formula: see text] be an element of a finite group [Formula: see text], and [Formula: see text] a prime factor of the order of [Formula: see text]. It is clear that there always exists a unique minimal subnormal subgroup containing [Formula: see text], say [Formula: see text]. We call the conjugacy class of [Formula: see text] in [Formula: see text] the sub-class of [Formula: see text] in [Formula: see text], see [G. Qian and Y. Yang, On sub-class sizes of finite groups, J. Aust. Math. Soc. (2020) 402–411]. In this paper, assume that [Formula: see text] is the product of the subgroups [Formula: see text] and [Formula: see text], we investigate the solvability, [Formula: see text]-nilpotence and supersolvability of the group [Formula: see text] under the condition that the sub-class sizes of prime power order elements in [Formula: see text] are [Formula: see text] free, [Formula: see text] free and square free, respectively, so that some known results relevant to conjugacy class sizes are generalized.

Optimum distance flag codes from spreads via perfect matchings in graphs

In this paper, we study flag codes on the vector space $${{\mathbb {F}}}_q^n$$ F q n , being q a prime power and $${{\mathbb {F}}}_q$$ F q the finite field of q elements. More precisely, we focus on flag codes that attain the maximum possible distance (optimum distance flag codes) and can be obtained from a spread of $${{\mathbb {F}}}_q^n$$ F q n . We characterize the set of admissible type vectors for this family of flag codes and also provide a construction of them based on well-known results about perfect matchings in graphs. This construction attains both the maximum distance for its type vector and the largest possible cardinality for that distance.

Upsilon Invariants from Cyclic Branched Covers

We extend the construction of Y-type invariants to null-homologous knots in rational homology three-spheres. By considering m-fold cyclic branched covers with m a prime power, this extension provides new knot concordance invariants of knots in S3. We give computations of some of these invariants for alternating knots and reprove independence results in the smooth concordance group.

K-Regular Matroids

<p>The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of near-regular matroids. Let k be a non-negative integer. This thesis considers the class of k-regular matroids, a generalization of the last two classes. Indeed, the classes of regular and near-regular matroids coincide with the classes of 0-regular and 1-regular matroids, respectively. This thesis extends many results for regular and near-regular matroids. In particular, for all k, the class of k-regular matroids is precisely the class of matroids representable over a particular partial field. Every 3-connected member of the classes of either regular or near-regular matroids has a unique representability property. This thesis extends this property to the 3-connected members of the class of k-regular matroids for all k. A matroid is [omega] -regular if it is k-regular for some k. It is shown that, for all k [greater than or equal to] 0, every 3-connected k-regular matroid is uniquely representable over the partial field canonically associated with the class of [omega] -regular matroids. To prove this result, the excluded-minor characterization of the class of k-regular matroids within the class of [omega] -regular matroids is first proved. It turns out that, for all k, there are a finite number of [omega] -regular excluded minors for the class of k-regular matroids. The proofs of the last two results on k-regular matroids are closely related. The result referred to next is quite different in this regard. The thesis determines, for all r and all k, the maximum number of points that a simple rank-r k-regular matroid can have and identifies all such matroids having this number. This last result generalizes the corresponding results for regular and near-regular matroids. Some of the main results for k-regular matroids are obtained via a matroid operation that is a generalization of the operation of [Delta] - Y exchange. This operation is called segment-cosegment exchange and, like the operation of [Delta] - Y exchange, has a dual operation. This thesis defines the generalized operation and its dual, and identifies many of their attractive properties. One property in particular, is that, for a partial field P, the set of excluded minors for representability over P is closed under the operations of segment-cosegment exchange and its dual. This result generalizes the corresponding result for [Delta] - Y and Y - [Delta] exchanges. Moreover, a consequence of it is that, for a prime power q, the number of excluded minors for GF(q)-representability is at least 2q-4.</p>

K-Regular Matroids

<p>The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of near-regular matroids. Let k be a non-negative integer. This thesis considers the class of k-regular matroids, a generalization of the last two classes. Indeed, the classes of regular and near-regular matroids coincide with the classes of 0-regular and 1-regular matroids, respectively. This thesis extends many results for regular and near-regular matroids. In particular, for all k, the class of k-regular matroids is precisely the class of matroids representable over a particular partial field. Every 3-connected member of the classes of either regular or near-regular matroids has a unique representability property. This thesis extends this property to the 3-connected members of the class of k-regular matroids for all k. A matroid is [omega] -regular if it is k-regular for some k. It is shown that, for all k [greater than or equal to] 0, every 3-connected k-regular matroid is uniquely representable over the partial field canonically associated with the class of [omega] -regular matroids. To prove this result, the excluded-minor characterization of the class of k-regular matroids within the class of [omega] -regular matroids is first proved. It turns out that, for all k, there are a finite number of [omega] -regular excluded minors for the class of k-regular matroids. The proofs of the last two results on k-regular matroids are closely related. The result referred to next is quite different in this regard. The thesis determines, for all r and all k, the maximum number of points that a simple rank-r k-regular matroid can have and identifies all such matroids having this number. This last result generalizes the corresponding results for regular and near-regular matroids. Some of the main results for k-regular matroids are obtained via a matroid operation that is a generalization of the operation of [Delta] - Y exchange. This operation is called segment-cosegment exchange and, like the operation of [Delta] - Y exchange, has a dual operation. This thesis defines the generalized operation and its dual, and identifies many of their attractive properties. One property in particular, is that, for a partial field P, the set of excluded minors for representability over P is closed under the operations of segment-cosegment exchange and its dual. This result generalizes the corresponding result for [Delta] - Y and Y - [Delta] exchanges. Moreover, a consequence of it is that, for a prime power q, the number of excluded minors for GF(q)-representability is at least 2q-4.</p>

On Baer cones in $$\mathrm {PG}(3, q)$$

Recently, in Innamorati and Zuanni (J. Geom 111:45, 2020. 10.1007/s00022-020-00557-0) the authors give a characterization of a Baer cone of $$\mathrm {PG}(3, q^2)$$ PG ( 3 , q 2 ) , q a prime power, as a subset of points of the projective space intersected by any line in at least one point and by every plane in $$q^2+1$$ q 2 + 1 , $$q^2+q+1$$ q 2 + q + 1 or $$q^3+q^2+1$$ q 3 + q 2 + 1 points. In this paper, we show that a similar characterization holds even without assuming that the order of the projective space is a square, and weakening the assumptions on the three intersection numbers with respect to the planes.