We consider additive codes over GF(4) that are self-dual with respect to the
Hermitian trace inner product. Such codes have a well-known interpretation as
quantum codes and correspond to isotropic systems. It has also been shown that
these codes can be represented as graphs, and that two codes are equivalent if
and only if the corresponding graphs are equivalent with respect to local
complementation and graph isomorphism. We use these facts to classify all codes
of length up to 12, where previously only all codes of length up to 9 were
known. We also classify all extremal Type II codes of length 14. Finally, we
find that the smallest Type I and Type II codes with trivial automorphism group
have length 9 and 12, respectively.Comment: 18 pages, 4 figure

Bachoc

Bouchet

Calderbank

Cannon

Danielsen

Gaborit

Hein

Huffman

Höhn

Lars Eirik Danielsen

Matthew G. Parker

McKay

Rains

Schlingemann

Sloane

Tonchev

Van den Nest

English

arXiv

Crossref

On the classification of all self-dual additive codes over GF(4) of length up to 12

We study objects that can be represented as graphs, error-correcting codes, quantum states, or Boolean functions. It is known that self-dual additive codes, which can also be interpreted as quantum states, can be represented as graphs, and that two codes are equivalent when the corresponding graphs are equivalent with respect to local complementation (LC). We give classifications of such codes. Circulant graph codes are introduced, and it is shown that some of these codes have highly regular graph representations. We show that the orbit of a bipartite graph under edge local complementation (ELC) corresponds to the equivalence class of a binary linear code. We classify ELC orbits, give a new method for classifying binary linear codes, and show that the information sets and the minimum distance of a code can be derived from the corresponding ELC orbit. Self-dual additive codes over GF(4) can also be interpreted as quadratic Boolean functions. In this context we define PARIHN, peak-to-average power ratio with respect to the {I, H, N}n transform, and prove that PARIHN equals the size of the maximum independent set over the associated LC orbit of graphs. We define the aperiodic propagation criteria (APC) of a Boolean function, and show that it is related to the minimum distance of a self-dual additive code over GF(4), and to the degree of entanglement in the associated quantum state. We give a generalization to non-quadratic Boolean functions, and relate APC to known cryptographic criteria. Interlace polynomials encode many properties of the LC and ELC orbits of a graph. We enumerate interlace polynomials and circle graphs, show that there exist non-unimodal interlace polynomials, and relate properties of interlace polynomials to properties of codes and quantum states. We define self-dual bent functions, and provide constructions and classifications

Danielsen, Lars Eirik

University of Bergen

On Connections Between Graphs, Codes, Quantum States, and Boolean Functions

AbstractWe consider additive codes over GF(4) that are self-dual with respect to the Hermitian trace inner product. Such codes have a well-known interpretation as quantum codes and correspond to isotropic systems. It has also been shown that these codes can be represented as graphs, and that two codes are equivalent if and only if the corresponding graphs are equivalent with respect to local complementation and graph isomorphism. We use these facts to classify all codes of length up to 12, where previously only all codes of length up to 9 were known. We also classify all extremal Type II codes of length 14. Finally, we find that the smallest Type I and Type II codes with trivial automorphism group have length 9 and 12, respectively

Parker, Matthew G.

Elsevier - Publisher Connector 

On the classification of all self-dual additive codes over GF(4) of length up to 12 

NORA - Norwegian Open Research Archives

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.103.7006

On the Classification of All Self-Dual Additive Codes over GF(4) of
  Length up to 12

On the Classification of All Self-Dual Additive Codes over GF(4) of Length up to 12

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University of Bergen

Elsevier - Publisher Connector

NORA - Norwegian Open Research Archives

Elsevier - Publisher Connector