Abstract
The bilinear form with associated identity matrix is used in coding theory to define the dual code of a linear code, also it endows linear codes with a metric space structure. This metric structure was studied for generalized toric codes and a characteristic decomposition was obtained, which led to several applications as the construction of stabilizer quantum codes and LCD codes. In this work, we use the study of bilinear forms over a finite field to give a decomposition of an arbitrary linear code similar to the one obtained for generalized toric codes. Such a decomposition, called the geometric decomposition of a linear code, can be obtained in a constructiveway; it allows us to express easily the dual code of a linear code and provides a method to construct stabilizer quantum codes, LCD codes and in some cases, a method to estimate their minimum distance. The proofs for characteristic 2 are different, but they are developed in parallel.
| Original language | English |
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| Title of host publication | Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics : Festschrift for Antonio Campillo on the Occasion of his 65th Birthday |
| Number of pages | 25 |
| Publisher | Springer |
| Publication date | 18 Sept 2018 |
| Pages | 537-561 |
| ISBN (Print) | 9783319968261 |
| ISBN (Electronic) | 9783319968278 |
| DOIs | |
| Publication status | Published - 18 Sept 2018 |