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70
Quantum Error Correction Via Codes Over GF(4)
, 1997
"... The problem of finding quantumerrorcorrecting codes is transformed into the problem of finding additive codes over the field GF(4) which are selforthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on s ..."
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Cited by 311 (21 self)
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The problem of finding quantumerrorcorrecting codes is transformed into the problem of finding additive codes over the field GF(4) which are selforthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.
A new upper bound on the minimal distance of selfdual codes
 IEEE Trans. Inform. Theory
, 1990
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The Shadow Theory of Modular and Unimodular Lattices
 J. Number Theory
"... It is shown that an ndimensional unimodular lattice has minimal norm at most 2[n=24]+2, unless n = 23 when the bound must be increased by 1. This result was previously known only for even unimodular lattices. Quebbemann had extended the bound for even unimodular lattices to strongly Nmodular even ..."
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Cited by 26 (6 self)
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It is shown that an ndimensional unimodular lattice has minimal norm at most 2[n=24]+2, unless n = 23 when the bound must be increased by 1. This result was previously known only for even unimodular lattices. Quebbemann had extended the bound for even unimodular lattices to strongly Nmodular even lattices for N in f1; 2; 3; 5; 6; 7; 11; 14; 15; 23g ; () and analogous bounds are established here for odd lattices satisfying certain technical conditions (which are trivial for N = 1 and 2). For N ? 1 in (), lattices meeting the new bound are constructed that are analogous to the "shorter" and "odd" Leech lattices. These include an odd associate of the 16dimensional BarnesWall lattice and shorter and odd associates of the CoxeterTodd lattice. A uniform construction is given for the (even) analogues of the Leech lattice, inspired by the fact that () is also the set of squarefree orders of elements of the Mathieu group M 23 . 1. Introduction The study of unimodular lattices (i.e. int...
Spectral Orbits and PeaktoAverage Power Ratio of Boolean Functions with respect to the {I, H, N}^n Transform
 SETA’04, SEQUENCES AND THEIR APPLICATIONS, SEOUL, ACCEPTED FOR PROCEEDINGS OF SETA04, LECTURE NOTES IN COMPUTER SCIENCE
, 2005
"... We enumerate the inequivalent selfdual additive codes over GF(4) of blocklength n, thereby extending the sequence A090899 in The OnLine Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a wellknown interpretation as quantum codes. They can also be represented by graphs, wh ..."
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Cited by 19 (15 self)
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We enumerate the inequivalent selfdual additive codes over GF(4) of blocklength n, thereby extending the sequence A090899 in The OnLine Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a wellknown interpretation as quantum codes. They can also be represented by graphs, where a simple graph operation generates the orbits of equivalent codes. We highlight the regularity and structure of some graphs that correspond to codes with high distance. The codes can also be interpreted as quadratic Boolean functions, where inequivalence takes on a spectral meaning. In this context we define PARIHN, peaktoaverage power ratio with respect to the {I, H, N} n transform set. We prove that PARIHN of a Boolean function is equivalent to the the size of the maximum independent set over the associated orbit of graphs. Finally we propose a construction technique to generate Boolean functions with low PARIHN and algebraic degree higher than 2.
Selfdual codes and invariant theory
 MATH. NACHRICHTEN
, 2006
"... There is a beautiful analogy between most of the notions for lattices and codes and it seems to be quite promising to develop coding theory analogues of concepts known in the theory of lattices and modular forms and vice versa. Some of these analogies are presented in this short note that intends to ..."
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Cited by 17 (8 self)
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There is a beautiful analogy between most of the notions for lattices and codes and it seems to be quite promising to develop coding theory analogues of concepts known in the theory of lattices and modular forms and vice versa. Some of these analogies are presented in this short note that intends to survey recent developments connected to my talk HeckeOperators for codes in Luminy, on May 9, 2007, where I introduce the KneserHeckeOperators mentioned in Section 3.5. More details can be found in the paper [7], a preprint of which is available on my homepage.
Designs And SelfDual Codes With Long Shadows
 J. COMBIN. THEORY SER. A
, 2002
"... In this paper we introduce the notion of sextremal codes for selfdual binary codes and we relate this notion to the existence of 1designs or 2designs in these codes. We extend the classification of codes with long shadows of [12] to codes with minimum distance 6, for which we give partial cl ..."
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Cited by 12 (0 self)
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In this paper we introduce the notion of sextremal codes for selfdual binary codes and we relate this notion to the existence of 1designs or 2designs in these codes. We extend the classification of codes with long shadows of [12] to codes with minimum distance 6, for which we give partial classification.
Directed Graph Representation of HalfRate Additive Codes over GF(4)
"... Abstract. We show that (n, 2 n, d) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on selfdual additive codes over GF(4), which correspond to undirected graphs. Graph representation greatly reduces the complexity of code classification, and enables ..."
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Cited by 7 (0 self)
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Abstract. We show that (n, 2 n, d) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on selfdual additive codes over GF(4), which correspond to undirected graphs. Graph representation greatly reduces the complexity of code classification, and enables us to classify additive (n, 2 n, d) codes over GF(4) of length up to 7. From this we also derive classifications of isodual and formally selfdual codes. We introduce new constructions of circulant and bordered circulant directed graph codes, and show that these codes will always be isodual. A computer search of all such codes of length up to 26 reveals that these constructions produce many codes of high minimum distance. In particular, we find new nearextremal formally selfdual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known selfdual codes. 1
An Improved Upper Bound on the Minimum Distance of DoublyEven SelfDual Codes
"... We derive a new upper bound on the minimum distance d of doublyeven selfdual codes of length n. Asymptotically, for n growing, it gives lim n!1 sup d=n (5 \Gamma 5 3=4 )=10 ! 0:165630, thus improving on the MallowsOdlyzkoSloane bound of 1=6 and our recent bound of 0.166315. Keywords: Selfdua ..."
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Cited by 6 (0 self)
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We derive a new upper bound on the minimum distance d of doublyeven selfdual codes of length n. Asymptotically, for n growing, it gives lim n!1 sup d=n (5 \Gamma 5 3=4 )=10 ! 0:165630, thus improving on the MallowsOdlyzkoSloane bound of 1=6 and our recent bound of 0.166315. Keywords: Selfdual codes, Distance distribution, Upper bounds. Running head: Bound on the distance of selfdual codes Corresponding author: S.Litsyn, tel.97236407286, fax.97236407095. 1. Introduction Selfdual codes attract a great deal of attention, mainly due to their intimate connections with improtant problems in algebra, combinatorics and number theory (see many references in [2, 3, 11, 14, 16]). A binary selfdual linear code C of length n and minimum distance d is doublyeven if all its weights are divisible by 4. By a result of Gleason (see e.g. [11, x19.2]) such codes exist only for n divisible by 8 (for a simple proof not based on invariant theory see [8]). Let d n be the maximum distance of a...
New asymptotic bounds for selfdual codes and lattices
 IEEE Trans. Inform. Theory
"... We give an independent proof of the KrasikovLitsyn bound d/n � (1−5 −1/4)/2 on doublyeven selfdual binary codes. The technique used (a refinement of the MallowsOdlyzkoSloane approach) extends easily to other families of selfdual codes, modular lattices, and quantum codes; in particular, we sho ..."
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We give an independent proof of the KrasikovLitsyn bound d/n � (1−5 −1/4)/2 on doublyeven selfdual binary codes. The technique used (a refinement of the MallowsOdlyzkoSloane approach) extends easily to other families of selfdual codes, modular lattices, and quantum codes; in particular, we show that the KrasikovLitsyn bound applies to singlyeven binary codes, and obtain an analogous bound for unimodular lattices. We also show that in each case, our bound differs from the true optimum by an amount growing faster than O ( √ n). 1