Quasi type IV codes over a non-unital ring
| Autor: | Alexis Bonnecaze, Alaa Altassan, Widyan Basaffar, Patrick Solé, Hatoon Shoaib, Adel Alahmadi |
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| Přispěvatelé: | King Abdulazziz University, Middle East Center of Algebra and its Applications (MECAA), King Abdulaziz University, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Algebraic structure
Computation 0102 computer and information sciences 02 engineering and technology Additive 4 -codes Type IV codes MSC(2010): Primary 94 B05 01 natural sciences Codes Combinatorics 0202 electrical engineering electronic engineering information engineering Canonical form Rings [MATH]Mathematics [math] Hamming weight Mathematics Secondary 16 A10 Algebra and Number Theory Type IV codes Mathematics Subject Classification Primary 94 B05 Applied Mathematics Local ring 020206 networking & telecommunications additive F 4 -codes Mass formula 010201 computation theory & mathematics [INFO.INFO-IT]Computer Science [cs]/Information Theory [cs.IT] Theory of computation Torsion (algebra) Mass formulas |
| Zdroj: | Applicable Algebra in Engineering, Communication and Computing Applicable Algebra in Engineering, Communication and Computing, Springer Verlag, 2021, ⟨10.1007/s00200-021-00488-6⟩ Applicable Algebra in Engineering, Communication and Computing, 2021 |
| ISSN: | 0938-1279 1432-0622 |
| DOI: | 10.1007/s00200-021-00488-6⟩ |
| Popis: | There is a local ring I of order 4, without identity for the multiplication, defined by generators and relations as $$\begin{aligned} I=\langle a,b \mid 2a=2b=0,\, a^{2}=b,\, \,ab=0 \rangle . \end{aligned}$$ We give a natural map between linear codes over I and additive codes over $${\mathbb{F}}_{4},$$ that allows for efficient computations. We study the algebraic structure of linear codes over this non-unital local ring, their generator and parity-check matrices. A canonical form for these matrices is given in the case of so-called nice codes. By analogy with $${\mathbb{Z}}_{4}$$ -codes, we define residue and torsion codes attached to a linear I-code. We introduce the notion of quasi self-dual codes (QSD) over I, and Type IV I-codes, that is, QSD codes all codewords of which have even Hamming weight. This is the natural analogue of Type IV codes over the field $${\mathbb{F}}_{4}.$$ Further, we define quasi Type IV codes over I as those QSD codes with an even torsion code. We give a mass formula for QSD codes, and another for quasi Type IV codes, and classify both types of codes, up to coordinate permutation equivalence, in short lengths. |
| Databáze: | OpenAIRE |
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