scholarly journals When the extension property does not hold

2017 ◽  
Vol 16 (05) ◽  
pp. 1750098
Author(s):  
Serhii Dyshko

A complete extension theorem for linear codes over a module alphabet, equipped with the symmetrized weight composition, is proven. It is also shown that the extension property with respect to a general weight function does not hold for module alphabets, that have a noncyclic socle.

2010 ◽  
Vol 77 (11) ◽  
pp. 1631-1643 ◽  
Author(s):  
M. Beghini ◽  
M. Benedetti ◽  
V. Fontanari ◽  
B.D. Monelli

2016 ◽  
Vol 15 (09) ◽  
pp. 1650162 ◽  
Author(s):  
Ali Assem

The extension problem for linear codes over modules with respect to Hamming weight was already settled in [J. A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc. 136 (2008) 699–706; Foundations of linear codes defined over finite modules: The extension theorem and MacWilliams identities, in Codes Over Rings, Series on Coding Theory and Cryptology, Vol. 6 (World Scientific, Singapore, 2009), pp. 124–190]. A similar problem arises naturally with respect to symmetrized weight compositions (SWC). In 2009, Wood proved that Frobenius bimodules have the extension property (EP) for SWC. More generally, in [N. ElGarem, N. Megahed and J. A. Wood, The extension theorem with respect to symmetrized weight compositions, in 4th Int. Castle Meeting on Coding Theory and Applications (2014)], it is shown that having a cyclic socle is sufficient for satisfying the property, while the necessity remained an open question. Here, landing in midway, a partial converse is proved. For a (not small) class of finite module alphabets, the cyclic socle is shown necessary to satisfy the EP. The idea is bridging to the case of Hamming weight through a new weight function. Note: All rings are finite with unity, and all modules are finite too. This may be re-emphasized in some statements. The convention for left homomorphisms is that inputs are to the left.


2007 ◽  
Vol 74 (4) ◽  
pp. 602-611 ◽  
Author(s):  
M. Beghini ◽  
L. Bertini ◽  
R. Di Lello ◽  
V. Fontanari

2020 ◽  
Vol 120 (1-2) ◽  
pp. 87-101
Author(s):  
Dario D. Monticelli ◽  
Fabio Punzo ◽  
Marco Squassina

We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems defined on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.


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