On the Classification of Hermitian Self-Dual Additive Codes Over ${\rm GF}(9)$

@article{Danielsen2012OnTC,
  title={On the Classification of Hermitian Self-Dual Additive Codes Over \$\{\rm GF\}(9)\$},
  author={L. Danielsen},
  journal={IEEE Transactions on Information Theory},
  year={2012},
  volume={58},
  pages={5500-5511}
}
  • L. Danielsen
  • Published 2012
  • Mathematics, Computer Science, Physics
  • IEEE Transactions on Information Theory
Additive codes over GF(9) that are self-dual with respect to the Hermitian trace inner product have a natural application in quantum information theory, where they correspond to ternary quantum error-correcting codes. However, these codes have so far received far less interest from coding theorists than self-dual additive codes over GF(4), which correspond to binary quantum codes. Self-dual additive codes over GF(9) have been classified up to length 8, and in this paper we extend the complete… Expand
Self-dual 𝔽q-linear 𝔽t-codes with an automorphism of prime order
TLDR
This work considers codes that are self-dual under one of these inner products and possess an automorphism of prime order, which is the natural generalization of the inner products used in [9, 30]. Expand
Searching for (near) Optimal Codes
TLDR
A new method of searching near optimal binary formally self-dual linear codes and additive codes from circulant graphs from circULant graphs is introduced. Expand
Enumeration and construction of additive cyclic codes over Galois rings
Let R = GR ( p ? , l ) be a Galois ring of characteristic p ? and cardinality p ? l , where p and l are prime integers. First, we give a canonical form decomposition for additive cyclic codes over RExpand
Additive codes over $GF(4)$ from circulant graphs
TLDR
This paper introduces a new method of searching (proposed) optimum additive codes from circulant graphs based on Danielsen and Parker's proved that every self-dual additive code over GF(4) is equivalent to a graph code. Expand
Repeated root cyclic 𝔽q-linear codes over 𝔽ql
TLDR
From the theory of linear codes over finite chain rings, enumeration, construction and encoder of these codes are investigated and the dual code of any cyclic F q -linear code over F p l of length n is studied. Expand
On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes
In [7], self-orthogonal additive codes over $\mathbb{F}_4$ under the trace inner product were connected to binary quantum codes; a similar connection was given in the nonbinary case in [33]. In thisExpand
Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity
TLDR
With the help of the weight enumerator machinery known from quantum error correction and the generalized shadow inequalities, new bounds are obtained on the existence of AME states in dimensions larger than two and the quantum MacWilliams identity is derived in the Bloch representation. Expand
Formally self-dual linear binary codes from circulant graphs
TLDR
A new method of searching (proposed) optimum formally self-duallinear binary codes from circulant graphs from circULant graphs is introduced. Expand
Hermitian Rank Metric Codes and Duality
TLDR
The concept of Hermitian linear complementary dual is introduced and it is proved that a self dual basis exists if and only if a rank metric codes endowed with a Hermitsian form is an odd integer. Expand
A new trace bilinear form on cyclic $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes
TLDR
According to this new trace bilinear form, bases and enumeration of cyclic $\Delta$-self-orthogonal and cyclic $ Delta$- self-dual $\mathbb{F}_q$-linear $\Mathbb {F}_{q^t}$-codes are investigated when $t=2$. Expand
...
1
2
...

References

SHOWING 1-10 OF 42 REFERENCES
On the classification of all self-dual additive codes over GF(4) of length up to 12
TLDR
All additive codes over GF(4) that are self-dual with respect to the Hermitian trace inner product are classified, where previously only all codes of length up to 9 were known. Expand
Graph-based classification of self-dual additive codes over finite fields
  • L. Danielsen
  • Mathematics, Computer Science
  • Adv. Math. Commun.
  • 2009
TLDR
It is proved that the minimum distance of a self-dual additive code is related to the minimum vertex degree in the associated graph orbit, and it is shown that some of these codes have highly regular graph representations. Expand
Euclidean and Hermitian self-dual MDS codes over large finite fields
TLDR
This paper builds many Euclidean and Hermitian self-dual MDS codes of length up to 12 over various finite fields GF(q), where q = 8, 9, 16, 25, 32, 41, 49, 53, 64, 81, and 128. Expand
Classification of Generalized Hadamard Matrices H(6,3) and Quaternary Hermitian Self-Dual Codes of Length 18
TLDR
It is shown that up to monomial equivalence, there are 85 generalized Hadamard matrices $H(6,3)$, and 245 inequivalent Hermitian self-dual codes of length 18 over $GF(4)$. Expand
Nonbinary Stabilizer Codes Over Finite Fields
TLDR
The basic theory of stabilizer codes over finite fields is described and a Galois theory for these objects is introduced, which generalizes the well-known notion of additive codes over F4 of the binary case. Expand
Quadratic Double Circulant Codes over Fields
  • P. Gaborit
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • 2002
TLDR
A generalization of the Pless symmetry codes to different fields is presented and it is proven that the automorphism group of some of these codes contains the group PSL2(q). Expand
Self-dual codes over the Kleinian four group
Abstract.We introduce self-dual codes over the Kleinian four group K=Z2×Z2 for a natural quadratic form on Kn and develop the theory. Topics studied are: weight enumerators, mass formulas,Expand
Nonbinary quantum codes
  • E. Rains
  • Mathematics, Physics
  • IEEE Trans. Inf. Theory
  • 1999
TLDR
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. Expand
New MDS or Near-MDS Self-Dual Codes
TLDR
It is shown that there exists a Euclidean self-dual MDS code of length n = q over GF(q) whenever q = 2m (m ges 2) using a Reed-Solomon (RS) code and its extension. Expand
On circulant self-dual codes over small fields
TLDR
With few exceptions, the codes achieve or improve the known lower bounds on the minimum distance of self-dual codes. Expand
...
1
2
3
4
5
...