ISSN:

1930-5346

eISSN:

1930-5338

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## Advances in Mathematics of Communications

November 2009 , Volume 3 , Issue 4

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2009, 3(4): 311-316
doi: 10.3934/amc.2009.3.311

*+*[Abstract](1258)*+*[PDF](111.5KB)**Abstract:**

There are extremal formally self-dual even codes which are not selfdual only for lengths 6, 10, 12, 14, 18, 20, 22, 28 and 30. The only length for which the classification has not been done yet is 30. In this note, new extremal formally self-dual even codes of length 30 are constructed using symmetric matrices.

2009, 3(4): 317-328
doi: 10.3934/amc.2009.3.317

*+*[Abstract](1406)*+*[PDF](195.8KB)**Abstract:**

We present the family of generalized AG convolutional codes, constructed by using algebraic geometric tools. This construction extends block generalized AG codes on the one hand and several algebraic constructions of convolutional codes on the other. The tools employed to define these codes are also used to obtain information about their parameters and to determine conditions such that the resulting codes have optimal free distance.

2009, 3(4): 329-348
doi: 10.3934/amc.2009.3.329

*+*[Abstract](1543)*+*[PDF](272.4KB)**Abstract:**

Quantum stabilizer states over F

_{m}can be represented as self-dual additive codes over F

_{m2}. These codes can be represented as weighted graphs, and orbits of graphs under the generalized local complementation operation correspond to equivalence classes of codes. We have previously used this fact to classify self-dual additive codes over F

_{4}. In this paper we classify selfdual additive codes over F

_{9}, F

_{16}, and F

_{25}. Assuming that the classical MDS conjecture holds, we are able to classify all self-dual additive MDS codes over F

_{9}by using an extension technique. We prove that the minimum distance of a self-dual additive code is related to the minimum vertex degree in the associated graph orbit. Circulant graph codes are introduced, and a computer search reveals that this set contains many strong codes. We show that some of these codes have highly regular graph representations.

2009, 3(4): 349-361
doi: 10.3934/amc.2009.3.349

*+*[Abstract](1361)*+*[PDF](194.6KB)**Abstract:**

In this paper, we are interested in the construction of maximum distance separable (MDS) self-dual codes over large prime fields that arise from the solutions of systems of diophantine equations. Using this method we con- struct many self-dualMDS (or near-MDS) codes of lengths up to 16 over various prime fields $GF(p)$, where $p$ = 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193 and 197. In addition, a number of optimal codes are presented for many lengths up to 40 over small prime fields $GF(p)$. Furthermore, our results on the minimum weight of self-dual codes over prime fields give a better bound than the Pless-Pierce bound obtained from a modified Gilbert-Varshamov bound.

2009, 3(4): 363-383
doi: 10.3934/amc.2009.3.363

*+*[Abstract](1754)*+*[PDF](283.1KB)**Abstract:**

The main aim of the

*classification*of linear codes is the evaluation of complete lists of representatives of the isometry classes. These classes are mostly defined with respect to

*linear*isometry, but it is well known that there is also the more general definition of

*semilinear*isometry taking the field automorphisms into account. This notion leads to bigger classes so the data becomes smaller. Hence we describe an algorithm that gives

*canonical*representatives of these bigger classes by calculating a unique generator matrix to a given linear code, in a well defined manner.

The algorithm is based on the partitioning and refinement idea which is also used to calculate the canonical labeling of a graph [12] and it similarly returns the automorphism group of the given linear code. The time needed by the implementation of the algorithm is comparable to Leon's program [10] for the calculation of the linear automorphism group of a linear code, but it additionally provides a unique representative and the automorphism group with respect to the more general notion of semilinear equivalence. The program can be used online under http://www.algorithm.uni-bayreuth.de/en/research/Coding_Theory/CanonicalForm/index.html.

2009, 3(4): 385-397
doi: 10.3934/amc.2009.3.385

*+*[Abstract](1188)*+*[PDF](190.0KB)**Abstract:**

We investigate partitions of the set $\mathbb F$

^{n}of all binary vectors of length $n$ into cosets of pairwise distinct linear Hamming codes (''non-parallel Hamming codes'') of length $n$. We present several constructions of partitions of $\mathbb F$

^{n}into non-parallel Hamming codes of length $n$ and discuss a lower bound on the number of different such partitions.

2009, 3(4): 399-408
doi: 10.3934/amc.2009.3.399

*+*[Abstract](1109)*+*[PDF](181.9KB)**Abstract:**

We introduce two types of curves of interest for coding theory purposes: the so-called Castle and weak Castle curves. We study the main properties of codes arising from these curves.

2009, 3(4): 409-420
doi: 10.3934/amc.2009.3.409

*+*[Abstract](1293)*+*[PDF](180.0KB)**Abstract:**

Repeated root Cyclic and Negacyclic codes over Galois rings have been studied much less than their simple root counterparts. This situation is beginning to change. For example, repeated root codes of length

*p*

^{s}, where $p$ is the characteristic of the alphabet ring, have been studied under some additional hypotheses. In each one of those cases, the ambient space for the codes has turned out to be a chain ring. In this paper, all remaining cases of cyclic and negacyclic codes of length

*p*

^{s}over a Galois ring alphabet are considered. In these cases the ambient space is a local ring with simple socle but not a chain ring. Nonetheless, by reducing the problem to one dealing with uniserial subambients, a method for computing the Hamming distance of these codes is provided.

2009, 3(4): 421-428
doi: 10.3934/amc.2009.3.421

*+*[Abstract](1114)*+*[PDF](146.7KB)**Abstract:**

A construction of

*2*

^{2n}-QAM sequences is given and an upper bound of the peak-to-mean envelope power ratio (PMEPR) is determined. Some former work can be viewed as special cases of this construction.

2009, 3(4): 429-430
doi: 10.3934/amc.2009.3.429

*+*[Abstract](979)*+*[PDF](44.6KB)**Abstract:**

Erratum to ''Edge number, minimum degree, maximum independent set, radius and diameter in twin-free graphs'' (Advances in Mathematics of Communications, Vol. 3, no. 1, 2009, 97–114).

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