Constacyclic codes over the ring Fq[u,v,w]/〈u2 − 1, v2 − 1,w3 − w, uv − vu,vw − wv,wu − uw〉

Gray Map ◽

Quantum Codes ◽

Constacyclic Codes ◽

Constacyclic Code ◽

Direct Sum ◽

Let [Formula: see text] be the ring [Formula: see text], where [Formula: see text] for any odd prime [Formula: see text] and positive integer [Formula: see text]. In this paper, we study constacyclic codes over the ring [Formula: see text]. We define a Gray map by a matrix and decompose a constacyclic code over the ring [Formula: see text] as the direct sum of constacyclic codes over [Formula: see text], we also characterize self-dual constacyclic codes over the ring [Formula: see text] and give necessary and sufficient conditions for constacyclic codes to be dual-containing. As an application, we give a method to construct quantum codes from dual-containing constacyclic codes over the ring [Formula: see text].

Quantum codes from a class of constacyclic codes over finite commutative rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500031 ◽

2019 ◽

Vol 19 (12) ◽

pp. 2150003 ◽

Cited By ~ 2

Author(s):

Hai Q. Dinh ◽

Tushar Bag ◽

Ashish K. Upadhyay ◽

Mohammad Ashraf ◽

Ghulam Mohammad ◽

...

Keyword(s):

Sufficient Conditions ◽

Commutative Rings ◽

Quantum Codes ◽

Constacyclic Codes ◽

Constacyclic Code ◽

Negacyclic Codes ◽

Direct Sum ◽

Direct Sum Decomposition ◽

Let [Formula: see text] be an odd prime, and [Formula: see text] be an integer such that [Formula: see text]. Using pairwise orthogonal idempotents [Formula: see text] of the ring [Formula: see text], with [Formula: see text], [Formula: see text] is decomposed as [Formula: see text], which contains the ring [Formula: see text] as a subring. It is shown that, for [Formula: see text], [Formula: see text], and it is invertible if and only if [Formula: see text] and [Formula: see text] are units of [Formula: see text]. In such cases, we study [Formula: see text]-constacyclic codes over [Formula: see text]. We present a direct sum decomposition of [Formula: see text]-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a [Formula: see text]-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over [Formula: see text], with better parameters than existing ones, are constructed from cyclic and negacyclic codes over [Formula: see text].

Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian Groups

Topological Algebra and its Applications ◽

10.1515/taa-2015-0007 ◽

2015 ◽

Vol 3 (1) ◽

Author(s):

Brendan Goldsmith ◽

Ketao Gong

Keyword(s):

Abelian Groups ◽

Sufficient Conditions ◽

Direct Sums ◽

Direct Sum ◽

Zero Entropy ◽

AbstractNecessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups.We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved.

Characterizations and Identities for Isosceles Triangular Numbers

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v14i2.3952 ◽

2021 ◽

Vol 14 (2) ◽

pp. 380-395

Author(s):

Jiramate Punpim ◽

Somphong Jitman

Keyword(s):

Positive Integer ◽

Sufficient Conditions ◽

Triangular Numbers ◽

Triangular Number ◽

Recursive Formulas ◽

Positive Integers ◽

Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

Partially Coherent Direct Sum Channels

Quantum ◽

10.22331/q-2021-07-15-504 ◽

2021 ◽

Vol 5 ◽

pp. 504

Author(s):

Stefano Chessa ◽

Vittorio Giovannetti

Keyword(s):

Sufficient Conditions ◽

Special Structure ◽

Quantum Channels ◽

Amplitude Damping ◽

Direct Sum ◽

Partially Coherent ◽

Quantum Capacity ◽

We introduce Partially Coherent Direct Sum (PCDS) quantum channels, as a generalization of the already known Direct Sum quantum channels. We derive necessary and sufficient conditions to identify the subset of those maps which are degradable, and provide a simplified expression for their quantum capacities. Interestingly, the special structure of PCDS allows us to extend the computation of the quantum capacity formula also for quantum channels which are explicitly not degradable (nor antidegradable). We show instances of applications of the results to dephasing channels, amplitude damping channels and combinations of the two.

On Generalizations of phi-2-Absorbing Primary Submodules

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v11i1.3126 ◽

2018 ◽

Vol 11 (1) ◽

pp. 35

Author(s):

Pairote Yiarayong ◽

Manoj Siripitukdet

Keyword(s):

Positive Integer ◽

Sufficient Conditions ◽

Commutative Rings ◽

Primary Submodules ◽

Primary Submodule ◽

Let $\phi: S(M) \rightarrow S(M) \cup \left\lbrace \emptyset\right\rbrace $ be a function where $S(M)$ is the set of all submodules of $M$. In this paper, we extend the concept of $\phi$-$2$-absorbing primary submodules to the context of $\phi$-$2$-absorbing semi-primary submodules. A proper submodule $N$ of $M$ is called a $\phi$-$2$-absorbing semi-primary submodule, if for each $m \in M$ and $a_{1}, a_{2}\in R$ with $a_{1}a_{2}m \in N - \phi(N)$, then $a_{1}a_{2}\in \sqrt{(N : M)}$ or $a_{1}m \in N$ or $a^{n}_{2}m\in N$, for some positive integer $n$. Those are extended from $2$-absorbing primary, weakly $2$-absorbing primary, almost $2$-absorbing primary, $\phi_{n}$-$2$-absorbing primary, $\omega$-$2$-absorbing primary and $\phi$-$2$-absorbing primary submodules, respectively. Some characterizations of $2$-absorbing semi-primary, $\phi_{n}$-$2$-absorbing semi-primary and $\phi$-$2$-absorbing semi-primary submodules are obtained. Moreover, we investigate relationships between $2$-absorbing semi-primary, $\phi_{n}$-$2$-absorbing semi-primary and $\phi$-primary submodules of modules over commutative rings. Finally, we obtain necessary and sufficient conditions of a $\phi$-$\phi$-$2$-absorbing semi-primary in order to be a $\phi$-$2$-absorbing semi-primary.

Eventually Pointed Principally Ordered Regular Semigroups

Sultan Qaboos University Journal for Science [SQUJS] ◽

10.24200/squjs.vol24iss2pp139-146 ◽

2020 ◽

Vol 24 (2) ◽

pp. 139

Author(s):

G.A. Pinto

Keyword(s):

Positive Integer ◽

Regular Semigroup ◽

Sufficient Conditions ◽

Regular Semigroups ◽

Completely Simple

An ordered regular semigroup, , is said to be principally ordered if for every there exists . A principally ordered regular semigroup is pointed if for every element, we have . Here we investigate those principally ordered regular semigroups that are eventually pointed in the sense that for all there exists a positive integer, , such that . Necessary and sufficient conditions for an eventually pointed principally ordered regular semigroup to be naturally ordered and to be completely simple are obtained. We describe the subalgebra of generated by a pair of comparable idempotents and such that .

On Sums of Progressions of Positive Integers

The Mathematical Gazette ◽

10.1017/s0025557200207109 ◽

1970 ◽

Vol 54 (388) ◽

pp. 113-115

Author(s):

R. L. Goodstein

Keyword(s):

Positive Integer ◽

Arithmetic Progression ◽

Sufficient Conditions ◽

The Common ◽

Positive Integers

We consider the problem of finding necessary and sufficient conditions for a positive integer to be the sum of an arithmetic progression of positive integers with a given common difference, starting with the case when the common difference is unity.

On a Theorem of Rav Concerning Egyptian Fractions

Canadian Mathematical Bulletin ◽

10.4153/cmb-1975-031-5 ◽

1975 ◽

Vol 18 (1) ◽

pp. 155-156 ◽

Cited By ~ 3

Author(s):

William A. Webb

Keyword(s):

Positive Integer ◽

Elementary Proof ◽

Sufficient Conditions ◽

Egyptian Fractions ◽

Image Position ◽

Complete Bibliography ◽

Problems involving Egyptian fractions (rationals whose numerator is 1 and whose denominator is a positive integer) have been extensively studied. (See [1] for a more complete bibliography). Some of the most interesting questions, many still unsolved, concern the solvability ofwhere k is fixed.In [2] Rav proved necessary and sufficient conditions for the solvabilty of the above equation, as a consequence of some other theorems which are rather complicated in their proofs. In this note we give a short, elementary proof of this theorem, and at the same time generalize it slightly.

On periodic rings

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171201001181 ◽

2001 ◽

Vol 25 (6) ◽

pp. 417-420

Author(s):

Xiankun Du ◽

Qi Yi

Keyword(s):

Sufficient Conditions ◽

Direct Sum ◽

Periodic Rings ◽

Nil Ring ◽

Positive Integers

It is proved that a ring is periodic if and only if, for any elementsxandy, there exist positive integersk,l,m, andnwith eitherk≠morl≠n, depending onxandy, for whichxkyl=xmyn. Necessary and sufficient conditions are established for a ring to be a direct sum of a nil ring and aJ-ring.

On the number of Kekulé structures of fluoranthene congeners

Journal of the Serbian Chemical Society ◽

10.2298/jsc091207077v ◽

2010 ◽

Vol 75 (8) ◽

pp. 1093-1098 ◽

Cited By ~ 4

Author(s):

Damir Vukicevic ◽

Jelena Djurdjevic ◽

Ivan Gutman

Keyword(s):

Positive Integer ◽

Sufficient Conditions ◽

Electron Systems ◽

Kekulé Structures ◽

Male And Female ◽

Structure Count ◽

The Kekul? structure count K of fluoranthene congeners is studied. It is shown that for such polycyclic conjugated ?-electron systems, either K = 0 or K ? 3. Moreover, for every t ? 3, there are infinitely many fluoranthene congeners having exactly t Kekul? structures. Three classes of Kekul?an fluoranthenes are distinguished: (i) ?0 - fluoranthene congeners in which neither the male nor the female benzenoid fragment has Kekul? structures, (ii) ?m - fluoranthene congeners in which the male benzenoid fragment has Kekul? structures, but the female does not, and (iii) ?fm - fluoranthene congeners in which both the male and female benzenoid fragments have Kekul? structures. Necessary and sufficient conditions are established for each class, ? = ?0, ?m, ?fm, such that for a given positive integer t, there exist fluoranthene congeners in ? with the property K = t.