348 research outputs found
The Dynamics of Group Codes: Dual Abelian Group Codes and Systems
Fundamental results concerning the dynamics of abelian group codes
(behaviors) and their duals are developed. Duals of sequence spaces over
locally compact abelian groups may be defined via Pontryagin duality; dual
group codes are orthogonal subgroups of dual sequence spaces. The dual of a
complete code or system is finite, and the dual of a Laurent code or system is
(anti-)Laurent. If C and C^\perp are dual codes, then the state spaces of C act
as the character groups of the state spaces of C^\perp. The controllability
properties of C are the observability properties of C^\perp. In particular, C
is (strongly) controllable if and only if C^\perp is (strongly) observable, and
the controller memory of C is the observer memory of C^\perp. The controller
granules of C act as the character groups of the observer granules of C^\perp.
Examples of minimal observer-form encoder and syndrome-former constructions are
given. Finally, every observer granule of C is an "end-around" controller
granule of C.Comment: 30 pages, 11 figures. To appear in IEEE Trans. Inform. Theory, 200
On the structure of framed vertex operator algebras and their pointwise frame stabilizers
In this paper, we study the structure of a general framed vertex operator
algebra. We show that the structure codes (C,D) of a framed VOA V satisfy
certain duality conditions. As a consequence, we prove that every framed VOA is
a simple current extension of the associated binary code VOA V_C. This result
would give a prospect on the classification of framed vertex operator algebras.
In addition, the pointwise frame stabilizer of V is studied. We completely
determine all automorphisms in this pointwise stabilizer, which are of order 1,
2 or 4. The 4A-twisted sector and the 4A-twisted orbifold theory of the famous
Moonshine VOA are also constructed explicitly. We verify that the top module of
this twisted sector is of dimension 1 and of weight 3/4 and the VOA obtained by
4A-twisted orbifold construction of the moonshine VOA is isomorphic to the
moonshine VOA itself.Comment: Version 3: 59 pages. Corrected version. 54 pages on my LaTeX system
version 2: We add Theorem 5.16 in which we give a necessary and sufficient
condtion for a code to be a structure code of a holomorphic framed VOA.
"hyperref" style is also introduce
MacWilliams Identities for Terminated Convolutional Codes
Shearer and McEliece [1977] showed that there is no MacWilliams identity for
the free distance spectra of orthogonal linear convolutional codes. We show
that on the other hand there does exist a MacWilliams identity between the
generating functions of the weight distributions per unit time of a linear
convolutional code C and its orthogonal code C^\perp, and that this
distribution is as useful as the free distance spectrum for estimating code
performance. These observations are similar to those made recently by
Bocharova, Hug, Johannesson and Kudryashov; however, we focus on terminating by
tail-biting rather than by truncation.Comment: 5 pages; accepted for 2010 IEEE International Symposium on
Information Theory, Austin, TX, June 13-1
Complete Weight Enumerators of Generalized Doubly-Even Self-Dual Codes
For any q which is a power of 2 we describe a finite subgroup of the group of
invertible complex q by q matrices under which the complete weight enumerators
of generalized doubly-even self-dual codes over the field with q elements are
invariant.
An explicit description of the invariant ring and some applications to
extremality of such codes are obtained in the case q=4
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