Hermitian codes are a class of very long algebraic-geometric (AG) codes constructed from Hermitian curves, which outperform Reed-Solomon codes defined over the same finite fields and with the same code rates, as recen...
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Hermitian codes are a class of very long algebraic-geometric (AG) codes constructed from Hermitian curves, which outperform Reed-Solomon codes defined over the same finite fields and with the same code rates, as recently demonstrated by the authors. However, since there are no soft-decision decoding algorithms for AG codes in the literature, the performance of Hermitian codes is limited and their potential is yet to be realized. An alternative method for achieving more significant coding gains is presented in this paper by serially concatenating long Hermitian codes with ring-trellis-coded modulation codes over the ring of integers Z(4) and evaluating their performance through simulation results on the AWGN and Rayleigh fading channels. The scheme achieves large coding gains over single Hermitian and Reed-Solomon codes with no increase in bandwidth use and a performance comparable with the well-known capacity-approaching codes. Copyright (C) 2007 John Wiley & Sons, Ltd.
In this correspondence, we describe a heuristic method for the construction of linear codes with given parameters n, k, q, and a prescribed minimum distance of at least d. Our approach is based on a function estimatin...
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In this correspondence, we describe a heuristic method for the construction of linear codes with given parameters n, k, q, and a prescribed minimum distance of at least d. Our approach is based on a function estimating the probability that a code of dimension k and blocklength n' < n over G F (q) is extendable to a code with the given properties. Combining this evaluation function with a search algorithm, we were able to improve 40 entries in the international tables for the best known minimum distance in the cases q = 2 5 71 9 and found at least two new optimal linear codes.
An n x n matrix A is said to be silver if, for i = 1, 2,..., n, each symbol in {1, 2,..., 2n-1} appears either in the ith row or the ith column of A. The 38th International Mathematical Olympiad asked whether a silver...
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An n x n matrix A is said to be silver if, for i = 1, 2,..., n, each symbol in {1, 2,..., 2n-1} appears either in the ith row or the ith column of A. The 38th International Mathematical Olympiad asked whether a silver matrix exists with n = 1997. More generally, a silver cube is a triple (K-n(d), I, c) where I is a maximum independent set in a Cartesian power of the complete graph K-n, and c : V (K-n(d)) -> {1, 2,..., d(n - 1) + 1} is a vertex colouring where, for v epsilon I, the closed neighbourhood N[v] sees every colour. Silver cubes are related to codes, dominating sets, and those with n a prime power are also related to finite geometry. We present here algebraic constructions, small examples, and a product construction. The nonexistence of silver cubes for d = 2 and some values of n, is proved using bounds from coding theory.
To protect sensitive security parameters in the non-volatile memory of integrated circuits, a device is designed that generates a special secret key (called IC-Eigenkey) to symmetrically encrypt this data. The IC-Eige...
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ISBN:
(纸本)9783540899938
To protect sensitive security parameters in the non-volatile memory of integrated circuits, a device is designed that generates a special secret key (called IC-Eigenkey) to symmetrically encrypt this data. The IC-Eigenkey is generated by the integrated circuit itself and therefore unknown to anybody else. The desired properties of such an IC-Eigenkey are postulated and a theoretical limit on the distribution of IC-Eigenkeys over an IC-production series is derived. The design of the IC-Eigenkey generator is based on silicon physical uncloneable functions. It exploits the marginal random variations of the propagation delays of gates and wires in an integrated circuit. A method is introduced that uses codewords of error control codes to configure the IC-Eigenkey generator in a way that the generated bits are as statistically independent of each other as possible.
Galois ring m-sequences were introduced in the late 1980s and early 1990s, and have near-optimal full periodic correlations. They are related to Z(4)-linear codes, and are used in CDMA communications. We consider peri...
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ISBN:
(纸本)9783540859116
Galois ring m-sequences were introduced in the late 1980s and early 1990s, and have near-optimal full periodic correlations. They are related to Z(4)-linear codes, and are used in CDMA communications. We consider periodic correlation and obtain algebraic expressions of the first two partial period correlation moments of the sequences belonging to families A, B and C. These correlation moments have applications in synchronisation performance of CDMA systems rising Galois ring sequences. The use of Association Schemes provides us with a new uniform technique for analyzing the sequence families A, B and C.
It is conjectured that quantum computers are able to solve certain problems more quickly than any deterministic or probabilistic computer. For instance, Shor’s algorithm is able to factor large integers in polynomial...
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It is conjectured that quantum computers are able to solve certain problems more quickly than any deterministic or probabilistic computer. For instance, Shor’s algorithm is able to factor large integers in polynomial time on a quantum computer. A quantum computer exploits the rules of quantum mechanics to speed up computations. However, it is a formidable task to build a quantum computer, since the quantum mechanical systems storing the information unavoidably interact with their environment. Therefore, one has to mitigate the resulting noise and decoherence effects to avoid computational errors. In this dissertation, I study various aspects of quantum error control codes – the key component of fault-tolerant quantum information processing. I present the fundamental theory and necessary background of quantum codes and construct many families of quantum block and convolutional codes over finite fields, in addition to families of subsystem codes. This dissertation is organized into three parts: Quantum Block Codes. After introducing the theory of quantum block codes, I establish conditions when BCH codes are self-orthogonal (or dual-containing) with respect to Euclidean and Hermitian inner products. In particular, I derive two families of nonbinary quantum BCH codes using the stabilizer formalism. I study duadic codes and establish the existence of families of degenerate quantum codes, as well as families of quantum codes derived from projective geometries. Subsystem Codes . Subsystem codes form a new class of quantum codes in which the underlying classical codes do not need to be self-orthogonal. I give an introduction to subsystem codes and present several methods for subsystem code constructions. I derive families of subsystem codes from classical BCH and RS codes and establish a family of optimal MDS subsystem codes. I establish propagation rules of subsystem codes and construct tables of upper and lower bounds on subsystem code parameters. Quantum Convolutional C
A method is given for the construction of linear codes with prescribed minimum distance and also prescribed minimum distance of the dual code. This works for codes over arbitrary finite fields. In the case of binary c...
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A method is given for the construction of linear codes with prescribed minimum distance and also prescribed minimum distance of the dual code. This works for codes over arbitrary finite fields. In the case of binary codes Matsumoto et al. showed how such codes can be used to construct cryptographic Boolean functions. This new method allows to compute new bounds on the size of such codes, extending the table of Matsumoto et al..
Our research explores the feasibility of using communication theory, error control (EC) coding theory specifically, for quantitatively modeling the protein translation initiation mechanism. The messenger RNA (mRNA) of...
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Our research explores the feasibility of using communication theory, error control (EC) coding theory specifically, for quantitatively modeling the protein translation initiation mechanism. The messenger RNA (mRNA) of Escherichia coli K-12 is modeled as a noisy (errored), encoded signal and the ribosome as a minimum Hamming distance decoder, where the 16S ribosomal RNA (rRNA) serves as a template for generating a set of valid codewords (the codebook). We tested the E. coli based coding models on 5' untranslated leader sequences of prokaryotic organisms of varying taxonomical relation to E. coli including: Salmonella typhimurium LT2, Bacillus subtilis, and Staphylococcus aureus Mu50. The model identified regions on the 5' untranslated leader where the minimum Hamming distance values of translated mRNA sub-sequences and non-translated genomic sequences differ the most. These regions correspond to the Shine-Dalgarno domain and the non-random domain. Applying the EC coding-based models to B. subtilis, and S. aureus Mu50 yielded results similar to those for E. coli K-12. Contrary to our expectations, the behavior of S. typhimurium LT2, the more taxonomically related to E. coli, resembled that of the non-translated sequence group. (C) 2004 Elsevier Ireland Ltd. All rights reserved.
In this paper, a coding-theory construction of Cartesian authentication codes is presented. The construction is a generalization of some known constructions. Within the framework of this generic construction, several ...
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In this paper, a coding-theory construction of Cartesian authentication codes is presented. The construction is a generalization of some known constructions. Within the framework of this generic construction, several classes of authentication codes using certain classes of error-correcting codes are described. The authentication codes presented in this paper are better than known ones with comparable parameters. It is demonstrated that the construction is related to certain combinatorial designs, such as difference matrices and generalized Hadamard matrices.
An explicit construction for nonbinary quantum Goppa codes exceeding the quantum Gilbert-Varshamov bound is given. First, we introduce a weighted symplectic inner product and show a method how to transform weighted co...
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An explicit construction for nonbinary quantum Goppa codes exceeding the quantum Gilbert-Varshamov bound is given. First, we introduce a weighted symplectic inner product and show a method how to transform weighted codes into quantum codes with respect to the standard symplectic inner product. Then an algorithm to construct a quantum code out of any hyperelliptic curve is presented and implemented in Magma. Finally, we apply a generalization of this algorithm to a tower of function fields by Stichtenoth and show that these codes lie above the quantum Gilbert-Varshamov bound.
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