127 research outputs found

    Solving Xq+1+X+a=0X^{q+1}+X+a=0 over Finite Fields

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    Solving the equation Pa(X):=Xq+1+X+a=0P_a(X):=X^{q+1}+X+a=0 over finite field \GF{Q}, where Q=pn,q=pkQ=p^n, q=p^k and pp is a prime, arises in many different contexts including finite geometry, the inverse Galois problem \cite{ACZ2000}, the construction of difference sets with Singer parameters \cite{DD2004}, determining cross-correlation between mm-sequences \cite{DOBBERTIN2006,HELLESETH2008} and to construct error-correcting codes \cite{Bracken2009}, as well as to speed up the index calculus method for computing discrete logarithms on finite fields \cite{GGGZ2013,GGGZ2013+} and on algebraic curves \cite{M2014}. Subsequently, in \cite{Bluher2004,HK2008,HK2010,BTT2014,Bluher2016,KM2019,CMPZ2019,MS2019}, the \GF{Q}-zeros of Pa(X)P_a(X) have been studied: in \cite{Bluher2004} it was shown that the possible values of the number of the zeros that Pa(X)P_a(X) has in \GF{Q} is 00, 11, 22 or pgcd(n,k)+1p^{\gcd(n, k)}+1. Some criteria for the number of the \GF{Q}-zeros of Pa(x)P_a(x) were found in \cite{HK2008,HK2010,BTT2014,KM2019,MS2019}. However, while the ultimate goal is to identify all the \GF{Q}-zeros, even in the case p=2p=2, it was solved only under the condition gcd(n,k)=1\gcd(n, k)=1 \cite{KM2019}. We discuss this equation without any restriction on pp and gcd(n,k)\gcd(n,k). New criteria for the number of the \GF{Q}-zeros of Pa(x)P_a(x) are proved. For the cases of one or two \GF{Q}-zeros, we provide explicit expressions for these rational zeros in terms of aa. For the case of pgcd(n,k)+1p^{\gcd(n, k)}+1 rational zeros, we provide a parametrization of such aa's and express the pgcd(n,k)+1p^{\gcd(n, k)}+1 rational zeros by using that parametrization

    Design and performance of CDMA codes for multiuser communications

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    Walsh and Gold sequences are fixed power codes and are widely used in multiuser CDMA communications. Their popularity is due to the ease of implementation. Availability of these code sets is limited because of their generating kernels. Emerging radio applications like sensor networks or multiple service types in mobile and peer-to-peer communications networks might benefit from flexibilities in code lengths and possible allocation methodologies provided by large set of code libraries. Walsh codes are linear phase and zero mean with unique number of zero crossings for each sequence within the set. DC sequence is part of the Walsh code set. Although these features are quite beneficial for source coding applications, they are not essential for spread spectrum communications. By relaxing these unnecessary constraints, new sets of orthogonal binary user codes (Walsh-like) for different lengths are obtained with comparable BER performance to standard code sets in all channel conditions. Although fixed power codes are easier to implement, mathematically speaking, varying power codes offer lower inter- and intra-code correlations. With recent advances in RF power amplifier design, it might be possible to implement multiple level orthogonal spread spectrum codes for an efficient direct sequence CDMA system. A number of multiple level integer codes have been generated by brute force search method for different lengths to highlight possible BER performance improvement over binary codes. An analytical design method has been developed for multiple level (variable power) spread spectrum codes using Karhunen-Loeve Transform (KLT) technique. Eigen decomposition technique is used to generate spread spectrum basis functions that are jointly spread in time and frequency domains for a given covariance matrix or power spectral density function. Since this is a closed form solution for orthogonal code set design, many options are possible for different code lengths. Design examples and performance simulations showed that spread spectrum KLT codes outperform or closely match with the standard codes employed in present CDMA systems. Hybrid (Kronecker) codes are generated by taking Kronecker product of two spreading code families in a two-stage orthogonal transmultiplexer structure and are judiciously allocated to users such that their inter-code correlations are minimized. It is shown that, BER performance of hybrid codes with a code selection and allocation algorithm is better than the performance of standard Walsh or Gold code sets for asynchronous CDMA communications. A redundant spreading code technique is proposed utilizing multiple stage orthogonal transmultiplexer structure where each user has its own pre-multiplexer. Each data bit is redundantly spread in the pre-multiplexer stage of a user with odd number of redundancy, and at the receiver, majority logic decision is employed on the detected redundant bits to obtain overall performance improvement. Simulation results showed that redundant spreading method improves BER performance significantly at low SNR channel conditions

    Estimating gate-set properties from random sequences

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    With quantum computing devices increasing in scale and complexity, there is a growing need for tools that obtain precise diagnostic information about quantum operations. However, current quantum devices are only capable of short unstructured gate sequences followed by native measurements. We accept this limitation and turn it into a new paradigm for characterizing quantum gate-sets. A single experiment - random sequence estimation - solves a wealth of estimation problems, with all complexity moved to classical post-processing. We derive robust channel variants of shadow estimation with close-to-optimal performance guarantees and use these as a primitive for partial, compressive and full process tomography as well as the learning of Pauli noise. We discuss applications to the quantum gate engineering cycle, and propose novel methods for the optimization of quantum gates and diagnosing cross-talk.Comment: 10+18 pages, two figures, substantially rewritten (made more intuitive, connected better to common experimental prescriptions, equipped with stronger numerical analysis
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