New quantum codes from self-dual codes over F_4

We present new constructions of binary quantum codes from quaternary linear Hermitian self-dual codes. Our main ingredients for these constructions are nearly self-orthogonal cyclic or duadic codes over F_4. An infinite family of $0$-dimensional binary quantum codes is provided. We give minimum distance lower bounds for our quantum codes in terms of the minimum distance of their ingredient linear codes. We also present new results on the minimum distance of linear cyclic codes using their fixed subcodes. Finally, we list many new record-breaking quantum codes obtained from our constructions.Comment: 16 page

On the equivalence of linear cyclic and constacyclic codes

We introduce new sufficient conditions for permutation and monomial equivalence of linear cyclic codes over various finite fields. We recall that monomial equivalence and isometric equivalence are the same relation for linear codes over finite fields. A necessary and sufficient condition for the monomial equivalence of linear cyclic codes through a shift map on their defining set is also given. Moreover, we provide new algebraic criteria for the monomial equivalence of constacyclic codes over $\mathbb{F}_4$. Finally, we prove that if $\gcd(3n,\phi(3n))=1$, then all permutation equivalent constacyclic codes of length $n$ over $\mathbb{F}_4$ are given by the action of multipliers. The results of this work allow us to prune the search algorithm for new linear codes and discover record-breaking linear and quantum codes.Comment: 18 page

Quantum stabilizer codes

We study quantum stabilizer codes and their connection to classical block codes. In addition, di erent constructions of quantum stabilizer codes and methods of modifying them are presented. Two-dimensional cyclic codes are recalled and a new method of obtaining quantum codes from 2-D cyclic codes is given. We also present a method of obtaining quantum stabilizer codes using additive codes over F4

Equivalence of constacyclic codes with shift constants of different orders

Let $a$ and $b$ be two non-zero elements of a finite field $\mathbb{F}_q$, where $q>2$. It has been shown that if $a$ and $b$ have the same multiplicative order in $\mathbb{F}_q$, then the families of $a$-constacyclic and $b$-constacyclic codes over $\mathbb{F}_q$ are monomially equivalent. In this paper, we investigate the monomial equivalence of $a$-constacyclic and $b$-constacyclic codes when $a$ and $b$ have distinct multiplicative orders. We present novel conditions for establishing monomial equivalence in such constacyclic codes, surpassing previous methods of determining monomially equivalent constacyclic and cyclic codes. As an application, we use these results to search for new linear codes more systematically. In particular, we present more than $70$ new record-breaking linear codes over various finite fields, as well as new binary quantum codes.Comment: 15 pages, 4 figures, 2 table

On the structure of primary ideals of a non-Laskerian group ring

In this paper, we study the structure of Rn = (Fp + uFp)[Zn; ] where u2 = 0 and (u) = −u. As a main result, we prove that this group ring is not Laskerian. Also, we classify the maximal ideals, prime ideals, and primary ideals and 2nd the ideals that have primary decomposition. We also nd J(Rn), Nil(Rn), Nil(Rn), and Nil(Rn) as additional results

Analysis of a subset selection scheme for wireless sensor networks in time-varying fading channels

One of the main challenges facing wireless sensor networks (WSNs) is the limited power resources available at small sensor nodes. It is therefore desired to reduce the power consumption of sensors while keeping the distortion between the source and its estimate at the fusion centre (FC) below a specific threshold. In this paper, we analyze a subset selection strategy to reduce the average transmission power of the WSN. We consider a two-hop network and assume the channels between the source and the relay sensors to be time-varying fading channels, modeled as Gilbert-Elliott channels. We show that when these channels are known at the FC, a subset of sensors can be selected by the FC to minimize transmit power while satisfying the distortion criterion. Through analysis, we derive the probability distribution of the size of this subset. We also consider practical aspects of implementing the proposed scheme, including channel estimation at relays. Through simulations, we compare the performance of the proposed scheme with schemes appearing in the literature. Simulation results confirm that for a certain range of end-to-end bit-error rates (BERs), the proposed scheme succeeds to achieve a superior power reduction compared to other schemes