arXiv subjects
Publications and source records attributed to T. Aaron Gulliver.
In this paper we investigate the structure of repeated root constacyclic codes of length $2^amp^r$ over $\mathbb{F}_{p^s}$ with $a\geq1$ and $(m,p)=1$. We characterize the codes in terms of their generator polynomials. This provides simple conditions on the existence of self-dual negacyclic codes. Further, we gave cases where the constacyclic codes are equivalent to cyclic codes.
In this paper we extend the work of Lisonek and Singh on construction X for quantum error-correcting codes to finite fields of order $p^2^ where p is prime. The results obtained are applied to the dual of Hermitian repeated root cyclic codes to generate new quantum error-correcting codes.
The definition of polar codes given by Arikan is explicit, but the construction complexity is an issue. This is due to the exponential growth in the size of the output alphabet of the bit-channels as the codeword length increases. Tal and Vardy recently presented a method for constructing polar codes which controls this growth. They approximated each bit-channel with a better channel and a worse channel while reducing the alphabet size. They constructed a polar code based on the worse channel and used the better channel to measure the distance from the optimal channel. This paper considers the knowledge gained from the perspective of the better channel. A method is presented using iterative upgrading of the bit-channels which successively results in a channel closer to the original one. It is shown that this approach can be used to obtain a channel arbitrarily close to the original channel, and therefore to the optimal construction of a polar code.
An algorithm is presented for approximating a single-user channel with a prime input alphabet size. The result is an upgraded version of the channel with a reduced output alphabet size. It is shown that this algorithm can be used to reduce the output alphabet size to the input alphabet size in most cases.
In a distributed storage systems (DSS), regenerating codes are used to optimize bandwidth in the repair process of a failed node. To optimize other DSS parameters such as computation and disk I/O, Distributed Replication-based Simple Storage (Dress) Codes consisting of an inner Fractional Repetition (FR) code and an outer MDS code are commonly used. Thus constructing FR codes is an important research problem, and several constructions using graphs and designs have been proposed. In this paper, we present an algorithm for constructing the node-packet distribution matrix of FR codes and thus enumerate some FR codes up to a given number of nodes n. We also present algorithms for constructing regular graphs which give rise to FR codes.
The Radio Environment Map (REM) provides an effective approach to Dynamic Spectrum Access (DSA) in Cognitive Radio Networks (CRNs). Previous results on REM construction show that there exists a tradeoff between the number of measurements (sensors) and REM accuracy. In this paper, we analyze this tradeoff and determine that the REM error is a decreasing and convex function of the number of measurements (sensors). The concept of geographic entropy is introduced to quantify this relationship. And the influence of sensor deployment on REM accuracy is examined using information theory techniques. The results obtained in this paper are applicable not only for the REM, but also for wireless sensor network deployment.
This paper considers cyclic DNA codes of arbitrary length over the ring $R=\F_2[u]/u^4-1$. A mapping is given between the elements of $R$ and the alphabet $\{A,C,G,T\}$ which allows the additive stem distance to be extended to this ring. Cyclic codes over $R$ are designed such that their images under the mapping are also cyclic or quasi-cyclic of index 2. The additive distance and hybridization energy are functions of the neighborhood energy.
In this paper, we consider the construction of linear lexicodes over finite chain rings by using a $B$-ordering over these rings and a selection criterion. % and a greedy Algorithm. As examples we give lexicodes over $\mathbb{Z}_4$ and $\mathbb{F}_2+u\mathbb{F}_2$. %First, greedy algorithms are presented to construct %lexicodes using a multiplicative property. Then, greedy algorithms %are given for the case when the selection criteria is not %multiplicative such as the minimum distance constraint. It is shown that this construction produces many optimal codes over rings and also good binary codes. Some of these codes meet the Gilbert bound. We also obtain optimal self-dual codes, in particular the octacode.
For $λ$ an $n$-th power of a unit in a finite chain ring we prove that $λ$-constacyclic repeated-root codes over some finite chain rings are equivalent to cyclic codes. This allows us to simplify the structure of some constacylic codes. We also study the $α+p β$-constacyclic codes of length $p^s$ over the Galois ring $GR(p^e,r)$.
We give the structure of $λ$-constacyclic codes of length $p^sm$ over $R=\mathbb{F}_{p^r}+u \mathbb{F}_{p^r}+...+ u^{e-1}\mathbb{F}_{p^r}$ with $λ\in \F_{p^r}^*$. We also give the structure of $λ$-constacyclic codes of length $p^sm$ with $λ=α_1+uα_2+...+u^{e-1} α_{e-1}$, where $α_1,α_2 \neq 0$ and study the self-duality of these codes.
New infinite families of quantum symmetric and asymmetric codes are constructed. Several of these are MDS. The codes obtained are shown to have parameters which are better than previously known. A number of known codes are special cases of the codes given here.
In this paper we give the structure of constacyclic codes over formal power series and chain rings. We also present necessary and sufficient conditions on the existence of MDS codes over principal ideal rings. These results allow for the construction of infinite families of MDS self-dual codes over finite chain rings, formal power series and principal ideal rings.
We construct codes over the ring $\mathbb{F}_2+u\mathbb{F}_2$ with $u^2=0$. These code are designed for use in DNA computing applications. The codes obtained satisfy the reverse complement constraint, the $GC$ content constraint and avoid the secondary structure. they are derived from the cyclic complement reversible codes over the ring $\mathbb{F}_2+u\mathbb{F}_2$. We also construct an infinite family of BCH DNA codes.
In this paper we investigate repeated root cyclic and negacyclic codes of length $p^rm$ over $\mathbb{F}_{p^s}$ with $(m,p)=1$. In the case $p$ odd, we give necessary and sufficient conditions on the existence of negacyclic self-dual codes. When $m=2m'$ with $m'$ odd, we characterize the codes in terms of their generator polynomials. This provides simple conditions on the existence of self-dual negacyclic codes, and generalizes the results of Dinh \cite{dinh}. We also answer an open problem concerning the number of self-dual cyclic codes given by Jia et al. \cite{jia}.
We determine the permutation groups that arise as the automorphism groups of cyclic combinatorial objects. As special cases we classify the automorphism groups of cyclic codes. We also give the permutations by which two cyclic combinatorial objects on $p^m$ elements are equivalent.
In this paper, we introduce a new quantum bit commitment protocol which is practically secure against entanglement attacks. A general cheating strategy is discussed and shown to be practically ineffective against the proposed approach.
This paper considers the use an entanglement breaking channel in the construction of a secure bit-commitment protocol. It is shown that this can be done via a depolarizing quantum channel.