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Infinite Families of Quantum-Classical Hybrid Codes

IEEE TRANSACTIONS ON INFORMATION THEORY 2021年第5期67卷 2847-2856页

作者： Nemec, Andrew Klappenecker, Andreas Texas A&M Univ Dept Comp Sci & Engn College Stn TX 77843 USA

Hybrid codes simultaneously encode both quantum and classical information into physical qubits. We give several general results about hybrid codes, most notably that the quantum codes comprising a genuine hybrid code must be impure and that hybrid codes can always detect more errors than comparable quantum codes. We also introduce the weight enumerators for general hybrid codes, which we then use to derive linear programming bounds. Finally, inspired by the construction of some families of nonadditive codes, we construct several infinite families of genuine hybrid codes with minimum distance two and three.

关键词： Codecs Measurement uncertainty Quantum channel Qubit Q measurement Indexes Quantum state Quantum error-correcting codes hybrid codes nonadditive codes codeword stabilized codes linear programming bound

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Simplicity conditions for binary orthogonal arrays

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DESIGNS CODES AND CRYPTOGRAPHY 2023年第1期91卷 151-163页

作者： Carlet, Claude Kiss, Rebeka Nagy, Gabor P. Univ Paris 08 Dept Math 2 Rue Liberte F-93526 St Denis France Univ Paris 08 F-93526 St Denis France Univ Paris 13 CNRS LAGA UMR 7539 F-93526 St Denis France Univ Bergen Dept Informat PB 7803 N-5020 Bergen Norway Univ Szeged Bolyai Inst Aradi Vertanuk Tere 1 H-6720 Szeged Hungary Budapest Univ Technol & Econ Dept Algebra Muegyet Rkp 3 H-1111 Budapest Hungary

It is known that correlation-immune (CI) Boolean functions used in the framework of side channel attacks need to have low Hamming weights. The supports of CI functions are (equivalently) simple orthogonal arrays, when their elements are written as rows of an array. The minimum Hamming weight of a CI function is then the same as the minimum number of rows in a simple orthogonal array. In this paper, we use Rao's bound to give a sufficient condition on the number of rows, for a binary orthogonal array (OA) to be simple. We apply this result for determining the minimum number of rows in all simple binary orthogonal arrays of strengths 2 and 3;we show that this minimum is the same in such case as for all OA, and we extend this observation to some OA of strengths 4 and 5. This allows us to reply positively, in the case of strengths 2 and 3, to a question raised by the first author and X. Chen on the monotonicity of the minimum Hamming weight of 2-CI Boolean functions, and to partially reply positively to the same question in the case of strengths 4 and 5.

关键词： Orthogonal array Correlation-immune Boolean function Rao's bound linear programming bound

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Unique optima of the Delsarte linear program

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DESIGNS CODES AND CRYPTOGRAPHY 2023年第6期91卷 2117-2140页

作者： Li, Rupert MIT Cambridge MA 02139 USA

The Delsarte linear program is used to bound the size of codes given their block length n and minimal distance d by taking a linear relaxation from codes to quasicodes. We study for which values of (n, d) this linear program has a unique optimum: while we show that it does not always have a unique optimum, we prove that it does if d > n/2 or if d <= 2. Introducing the Krawtchouk decomposition of a quasicode, we prove there exist optima to the (n, 2e) and (n - 1, 2e - 1) linear programs that have essentially identical Krawtchouk decompositions, revealing a parity phenomenon among the Delsarte linear programs. We generalize the notion of extending and puncturing codes to quasicodes, from which we see that this parity relationship is given by extending/puncturing. We further characterize these pairs of optima, in particular demonstrating that they exhibit a symmetry property, effectively halving the number of decision variables.

关键词： Delsarte linear program linear programming bound Codes Coding theory Quasicode Krawtchouk polynomials

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bounds for binary codes with narrow distance distributions

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IEEE TRANSACTIONS ON INFORMATION THEORY 2007年第8期53卷 2760-2768页

作者： Roth, Ron M. Seroussi, Gadiel Technion Israel Inst Technol Dept Comp Sci IL-32000 Haifa Israel Hewlett Packard Labs Palo Alto CA 94303 USA Univ Republica Montevideo Uruguay

New lower bounds are presented on the second moment of the distance distribution of binary codes, in terms of the first moment of the distribution. These bounds are used to obtain upper bounds on the size of codes whose maximum distance is close to their minimum distance. It is then demonstrated how such bounds can be applied to bound from below the smallest attainable ratio between the maximum distance and the minimum distance of codes. Finally, counterparts of the bounds are derived for the special case of constant-weight codes.

关键词： constant-weight codes distance distribution equidistant codes Grey-Rankin bound linear programming bound quasi-symmetric designs self-complementary codes

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Improved upper bounds on sizes of codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2002年第4期48卷 880-886页

作者： Mounits, B Etzion, T Litsyn, S Technion Israel Inst Technol Dept Math IL-32000 Haifa Israel Technion Israel Inst Technol Dept Comp Sci IL-32000 Haifa Israel Tel Aviv Univ Dept Elect Engn Syst IL-69978 Tel Aviv Israel

Let A(n, d) denote the maximum possible number of codewords in a binary code of length n and minimum Hamming distance d. For large values of n, the best known upper bound, for fixed d, is the Johnson bound. We give a new upper bound which is at least as good as the Johnson bound for all values of n and d, and for each d there are infinitely many values of n for which the new bound is better than the Johnson bound. For small values of n and d, the best known method to obtain upper bounds on A(n, d) is linear programming. We give new inequalities for the linear programming and show that with these new inequalities some of the known bounds on A(n, d) for n less than or equal to 28 are improved.

关键词： A(n, d) holes Johnson bound linear programming bound

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Computing approximate solutions of the maximum covering problem with GRASP

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JOURNAL OF HEURISTICS 1998年第2期4卷 161-177页

作者： Resende, MGC AT&T Labs Res Florham Park NJ 07932 USA

We consider the maximum covering problem, a combinatorial optimization problem that arises in many facility location problems. In this problem, a potential facility site covers a set of demand points. With each demand point, we associate a nonnegative weight. The task is to select a subset of p > 0 sites from the set of potential facility sites, such that the sum of weights of the covered demand points is maximized. We describe a greedy randomized adaptive search procedure (GRASP) for the maximum covering problem that finds good, though not necessarily optimum, placement configurations. We describe a well-known upper bound on the maximum coverage which can be computed by solving a linear program and show that on large instances, the GRASP can produce facility placements that are nearly optimal.

关键词： maximum covering problem facility location heuristic GRASP linear programming bound

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New minimum distance bounds for linear codes over GF(5)

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DISCRETE MATHEMATICS 2004年第1-3期275卷 97-110页

作者： Daskalov, R Hristov, P Metodieva, E Tec Univ Gabrovo Dept Math Gabrovo 5300 Bulgaria

Let [n, k, d](q)-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). In this paper, 32 new codes over GF(5) are constructed and the nonexistence of 51 codes is proved. (C) 2003 El... 详细信息

关键词： linear codes over GF(5) quasi-cyclic codes quasi-twisted codes linear programming bound

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bounds on mixed binary/ternary codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 1998年第1期44卷 140-161页

作者： Brouwer, AE Hamalainen, HO Ostergard, PRJ Sloane, NJA Eindhoven Univ Technol Dept Math NL-5600 MB Eindhoven Netherlands Helsinki Univ Technol Dept Comp Sci & Engn HUT 02015 Finland AT&T Bell Labs Res Informat Sci Res Ctr Florham Park NJ 07932 USA

Upper and lower bounds are presented for the maximal possible size of mixed binary/ternary error-correcting codes. A table up to length 13 is included, The upper bounds are obtained by applying the linear programming ... 详细信息

关键词： binary codes clique finding linear programming bound mixed codes tabu search ternary codes

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New explicit bounds for ordered codes and (t, m, s)-nets

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DISCRETE MATHEMATICS 2010年第4期310卷 970-975页

作者： Trinker, Horst Salzburg Univ Dept Math A-5020 Salzburg Austria

We derive two explicit bounds from the linear programming bound for ordered codes and ordered orthogonal arrays. While ordered codes generalize the concept of error-correcting block codes in Hamming space, ordered orthogonal arrays play an important role in the context of numerical integration and quasi-Monte Carlo methods because of their equivalence to (t, m, s)-nets, low-discrepancy point sets in the s-dimensional unit cube whenever t is reasonably small. The first bound we prove is a refinement of the Plotkin bound;the second bound shares its parameter range with the quadratic bound by Bierbrauer as well as the Plotkin bound. Both bounds yield improvements for various parameters. (c) 2009 Elsevier B.V. All rights reserved.

关键词： linear programming bound Ordered codes (t, m, s)-nets Ordered orthogonal arrays

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On the Algebraic Combinatorics of Injections and its Applications to Injection Codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2020年第11期66卷 6898-6907页

作者： Dukes, Peter J. Ihringer, Ferdinand Lindzey, Nathan Univ Victoria Dept Math & Stat Victoria BC V8P 5C2 Canada Univ Ghent Dept Math Anal Log & Discrete Math B-9000 Ghent Belgium Univ Colorado Dept Comp Sci Boulder CO 80309 USA

We consider the algebraic combinatorics of the set of injections from a k-element set to an n-element set. In particular, we give a new combinatorial formula for the spherical functions of the Gelfand pair (S-k x S-n, diag(S-k) x Sn- k). We use this combinatorial formula to give new Delsarte linear programming bounds on the size of codes over injections.

关键词： Standards Shape Strips Hamming distance linear programming Eigenvalues and eigenfunctions Indexes Permutation code linear programming bound injection partial permutation representation theory association scheme

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