arXiv subjects
Publications and source records attributed to Hopein Christofen Tang.
In this work, we present novel approaches for constructing linear codes over $\ZZ_4$ from the known ones. We succeeded in obtaining new linear codes, many of which are optimal. In particular, we found all optimal codes for $k_1=2,~k_2=0$ and many optimal codes for $k_1=3,~k_2=0.$
We interpret the symmetrized weight enumerator of linear codes over finite commutative Frobenius rings as a summation over multisets and thereby provide a new proof of the MacWilliams identity for the symmetrized weight enumerator. The proof and the identity are expressed in combinatorial terms that do not require generating characters. We also generalize the symmetrized weight enumerator with respect to supports and codeword tuples, and our multiset approach enables us to derive new and general MacWilliams identities expressed in combinatorial terms.
We obtain all possible parameters of Plotkin-optimal two-Lee weight projective codes over $\mathbb{Z}_4,$ together with their weight distributions. We show the existence of codes with these parameters as well as their weight distributions by constructing an infinite family of two-weight codes. Previously known codes constructed by Shi et al. (\emph{Des Codes Cryptogr.} {\bf 88}(3):1-13, 2020) can be derived as a special case of our results. We also prove that the Gray image of any Plotkin-optimal two-Lee weight projective codes over $\mathbb{Z}_4$ has the same parameters and weight distribution as some two-weight binary projective codes of type SU1 in the sense of Calderbank and Kantor (\emph{Bull. Lond. Math. Soc.} {\bf 18}:97-122, 1986).
In this work, we study a class of skew cyclic codes over the ring $R:=\mathbb{Z}_4+v\mathbb{Z}_4,$ where $v^2=v,$ with an automorphism $θ$ and a derivation $Δ_θ,$ namely codes as modules over a skew polynomial ring $R[x;θ,Δ_θ],$ whose multiplication is defined using an automorphism $θ$ and a derivation $Δ_θ.$ We investigate the structures of a skew polynomial ring $R[x;θ,Δ_θ].$ We define $Δ_θ$-cyclic codes as a generalization of the notion of cyclic codes. The properties of $Δ_θ$-cyclic codes as well as dual $Δ_θ$-cyclic codes are derived. As an application, some new linear codes over $\mathbb{Z}_4$ with good parameters are obtained by Plotkin sum construction, also via a Gray map as well as residue and torsion codes of these codes.
In this work, we introduce the harmonic generalization of the $m$-tuple weight enumerators of codes over finite Frobenius rings. A harmonic version of the MacWilliams-type identity for $m$-tuple weight enumerators of codes over finite Frobenius ring is also given. Moreover, we define the demi-matroid analogue of well-known polynomials from matroid theory, namely Tutte polynomials and coboundary polynomials, and associate them with a harmonic function. We also prove the Greene-type identity relating these polynomials to the harmonic $m$-tuple weight enumerators of codes over finite Frobenius rings. As an application of this Greene-type identity, we provide a simple combinatorial proof of the MacWilliams-type identity for harmonic $m$-tuple weight enumerators over finite Frobenius rings. Finally, we provide the structure of the relative invariant spaces containing the harmonic $m$-tuple weight enumerators of self-dual codes over finite fields.
We present new values and bounds on the (normalised) closeness centrality $\bar{\mathsf{C}}_C$ of connected graphs and on its product $\bar{l}\bar{\mathsf{C}}_C$ with the mean distance $\bar{l}$ of these graphs. Our main result presents the fundamental bounds $1\leq \bar{l}\bar{\mathsf{C}}_C<2$. The lower bound is tight and the upper bound is asymptotically tight. Combining the lower bound with known upper bounds on the mean distance, we find ten new lower bounds for the closeness centrality of graphs. We also present explicit expressions for $\bar{\mathsf{C}}_C$ and $\bar{l}\bar{\mathsf{C}}_C$ for specific families of graphs. Elegantly and perhaps surprisingly, the asymptotic values $n\bar{\mathsf{C}}_C\big(P_n\big)$ and of $n\bar{\mathsf{C}}_C\big(L_n\big)$ both equal $π$, and the asymptotic limits of $\bar{l}\bar{\mathsf{C}}_C$ for these families of graphs are both equal to $π/3$. We conjecture that the set of values $\bar{l}\bar{\mathsf{C}}_C$ for all connected graphs is dense in the interval $[1,2)$.
In 2003, Deutsch and Elizalde defined a family of bijective maps between the set of Dyck paths to itself which is induced by some particular permutations. In this paper, we extend the construction of the maps by allowing the permutation to be arbitrary. We characterise the permutations which generate the same map and find all permutations generating a bijection among Dyck paths. Consequently, we give a new combinatorial interpretation of the quantity $(2n-1)!!$ as well as some new statistics of Dyck paths which are equidistributed to some known height statistics via our generalised maps.
In this article, we investigate properties of cyclic codes over a finite non-chain ring $\mathbb{F}_q+v\mathbb{F}_q+v^2\mathbb{F}_q+v^3\mathbb{F}_q+v^4\mathbb{F}_q,$ where $q=p^r,$ $r$ is a positive integer, $p$ is an odd prime, $4 \mid (p-1),$ and $v^5=v.$ As an application, we construct several quantum error correcting codes over the finite field $\mathbb{F}_q.$