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International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings., 2004
We construct maximum distance separable quantum error-correcting codes. The codes are defined over q-dimensional quantum systems, where q is any prime power. The construction yields quantum MDS codes of length up to q+1 for all possible dimensions and some quantum MDS codes of length up to q2+1.
Rötteler, M., Grassl, M., Beth, T.
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We construct maximum distance separable quantum error-correcting codes. The codes are defined over q-dimensional quantum systems, where q is any prime power. The construction yields quantum MDS codes of length up to q+1 for all possible dimensions and some quantum MDS codes of length up to q2+1.
Rötteler, M., Grassl, M., Beth, T.
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IEEE Transactions on Information Theory, 1983
Summary: Maximum distance separable (MDS) convolutional codes are defined as the row space over \(F(D)\) of totally nonsingular polynomial matrices in the indeterminate \(D\). These codes may be used to transmit information on \(n\) parallel channels when a temporary or even an infinite break can occur in some of these channels.
Philippe Piret, Thijs Krol
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Summary: Maximum distance separable (MDS) convolutional codes are defined as the row space over \(F(D)\) of totally nonsingular polynomial matrices in the indeterminate \(D\). These codes may be used to transmit information on \(n\) parallel channels when a temporary or even an infinite break can occur in some of these channels.
Philippe Piret, Thijs Krol
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Annals of Combinatorics, 2005
An MDS code over an alphabet of size \(q\) is a set of length \(n\) vectors with \(q^k\) elements where the minimum distance \(d\) satisfies \(d=n-k+1.\) Any MDS code satisfies the bound \(n \leq q+k-1.\) If equality is met in this bound the code is said to be of maximal length.
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An MDS code over an alphabet of size \(q\) is a set of length \(n\) vectors with \(q^k\) elements where the minimum distance \(d\) satisfies \(d=n-k+1.\) Any MDS code satisfies the bound \(n \leq q+k-1.\) If equality is met in this bound the code is said to be of maximal length.
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International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings., 2004
In this paper we develop a complete generalization of the building-up method [J.-L. Kim, (2001)] for the Euclidean and Hermitian self-dual codes over finite fields GF(q). Using this method we construct many new Euclidean and Hermitian self-dual MDS (or near MDS) codes of length up to 12 over various finite fields GF(q), where q=8, 9, 16, 25, 32, 41, 49,
Jon-Lark Kim, Yoonjin Lee
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In this paper we develop a complete generalization of the building-up method [J.-L. Kim, (2001)] for the Euclidean and Hermitian self-dual codes over finite fields GF(q). Using this method we construct many new Euclidean and Hermitian self-dual MDS (or near MDS) codes of length up to 12 over various finite fields GF(q), where q=8, 9, 16, 25, 32, 41, 49,
Jon-Lark Kim, Yoonjin Lee
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Finite Fields and Their Applications, 2022
For a linear code of parameters \([n, k, d],\) the Singleton bound says that \(d \leq n-k+1.\) If this equality holds, i.e., \(d =n-k+1,\) then the code is called a maximum distance separable (MDS) code. If \(d =n-k,\) then the code is called an almost MDS (AMDS for short) code.
Xiaojun Geng +3 more
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For a linear code of parameters \([n, k, d],\) the Singleton bound says that \(d \leq n-k+1.\) If this equality holds, i.e., \(d =n-k+1,\) then the code is called a maximum distance separable (MDS) code. If \(d =n-k,\) then the code is called an almost MDS (AMDS for short) code.
Xiaojun Geng +3 more
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2000
We investigate the question when a cyclic code is maximum distance separable (MDS). For codes of (co-)dimension 3, this question is related to permutation properties of the polynomial (x b -1)/(x-1) for a certain b. Using results on these polynomials we prove that over fields of odd characteristic the only MDS cyclic codes of dimension 3 are the Reed ...
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We investigate the question when a cyclic code is maximum distance separable (MDS). For codes of (co-)dimension 3, this question is related to permutation properties of the polynomial (x b -1)/(x-1) for a certain b. Using results on these polynomials we prove that over fields of odd characteristic the only MDS cyclic codes of dimension 3 are the Reed ...
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Journal of Geometry, 1993
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Karzel, Helmut, Maxson, Carl J.
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zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Karzel, Helmut, Maxson, Carl J.
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On the Stopping Redundancy of MDS Codes
2006 IEEE International Symposium on Information Theory, 2006The stopping redundancy of a linear code is defined as the minimum number of rows in its parity-check matrix such that the smallest stopping sets have size equal to the minimum distance of the code. We derive new upper bounds on the stopping redundancy of maximum distance separable (MDS) codes, and show how they improve upon previously known results ...
Junsheng Han, Paul H. Siegel
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IEEE Transactions on Information Theory, 1999
Summary: Let \(\mathbb{F}_q\) denote the finite field \(GF(q)\) and let \(b\) be a positive integer. MDS codes over the symbol alphabet \(\mathbb{F}^b_q\) are considered that are linear over \(\mathbb{F}_q\) and have sparse (``low-density'') parity-check and generator matrices over \(\mathbb{F}_q\) that are systematic over \(\mathbb{F}_q^b\).
Mario Blaum, Ron M. Roth
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Summary: Let \(\mathbb{F}_q\) denote the finite field \(GF(q)\) and let \(b\) be a positive integer. MDS codes over the symbol alphabet \(\mathbb{F}^b_q\) are considered that are linear over \(\mathbb{F}_q\) and have sparse (``low-density'') parity-check and generator matrices over \(\mathbb{F}_q\) that are systematic over \(\mathbb{F}_q^b\).
Mario Blaum, Ron M. Roth
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On the Covering Radius of MDS Codes
IEEE Transactions on Information Theory, 2015For a linear maximum distance separable (MDS) code with redundancy $r$ , the covering radius is either $r$ or $r-1$ . However, for $r>3$ , few examples of $q$ -ary linear MDS codes with radius $r-1$ are known, including the Reed–Solomon codes with length $q+1$ . In this paper, for redundancies $r$ as large as
BARTOLI, DANIELE +2 more
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