Applications of Coding Theory to the Construction of Modular Lattices
TLDR
Using invariant theory, a basis is given for the space of invariants to which belongs the three variables weight enumerator of a self-dual code, and a general bound for the weight of such codes is derived.About:
This article is published in Journal of Combinatorial Theory, Series A.The article was published on 1997-04-01 and is currently open access. It has received 172 citations till now. The article focuses on the topics: Linear code & Block code.read more
Citations
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Nonbinary quantum codes
TL;DR: In this article, it was shown that the Calderbank-Shor-Steane codes and GF(4)-linear codes are special cases of the same construction, and that they can be used to construct families of quantum codes from certain codes over number fields.
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Self-Dual Codes
Eric M. Rains,Neil J. A. Sloane +1 more
TL;DR: A survey of self-dual codes is given in this paper, where the authors present a comprehensive bibliography of codes over F2, F3, F4, Fq, Z4, Zm, shadow codes, weight enumerators, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes.
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Nonbinary quantum codes
TL;DR: This work considers codes derived from finite symplectic geometry assumed to have additional global symmetries, and gets analogs of quadratic residue codes, including a single-error-correcting code encoding one letter in five, for any alphabet size.
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Cyclic codes and self-dual codes over F/sub 2/+uF/sub 2/
Alexis Bonnecaze,P. Udaya +1 more
TL;DR: In this article, linear cyclic codes over the ring F/sub 2 + u/sup 2/=0 were introduced and studied by analogy with the Z/sub 4/ case.
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Type II codes over F/sub 2/+uF/sub 2/
TL;DR: In this paper, the alphabet F/sub 2/uF/sub/2/1/2 is viewed as a quotient of the Gaussian integers by the ideal (2), and self-dual codes with Lee weights a multiple of 4 are called Type II.
References
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Book
The Theory of Error-Correcting Codes
TL;DR: This book presents an introduction to BCH Codes and Finite Fields, and methods for Combining Codes, and discusses self-dual Codes and Invariant Theory, as well as nonlinear Codes, Hadamard Matrices, Designs and the Golay Code.
Book
Sphere packings, lattices, and groups
TL;DR: The second edition of this book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space?
Journal ArticleDOI
New 5-designs
E. F. Assmus,H. F. Mattson +1 more
TL;DR: In this article, the theory of error-correcting codes has been used to construct new 5-designs on 24 and 48 points as well as the two classic 5 designs on 12 and 24 points associated with the Mathieu groups M12 and M24.
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Self-dual codes over the integers modulo 4
John H. Conway,Neil J. A. Sloane +1 more
TL;DR: This paper deduces analogues theorems for the “symmetrized” weight enumerator (which ignores the difference between +1 and −1 coordinates) and the Hamming weight enumerATOR and classify those self-orthogonal codes that are generated by words of type ±14On−4.