Quasi type IV codes over a non-unital ring

There is a local ring I of order 4,  without identity for the multiplication, defined by generators and relations as I = ⟨ a , b ∣ 2 a = 2 b = 0 , a 2 = b , a b = 0 ⟩ . We give a natural map between linear codes over I and additive codes over F 4 , that allows for efficient computations. We study th...

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Published inApplicable algebra in engineering, communication and computing Vol. 32; no. 3; pp. 217 - 228
Main Authors Alahmadi, Adel, Altassan, Alaa, Basaffar, Widyan, Bonnecaze, Alexis, Shoaib, Hatoon, Solé, Patrick
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2021
Springer Nature B.V
Springer Verlag
Subjects
Artificial Intelligence
Binary system
Canonical forms
Codes
Computer Hardware
Computer Science
Information Theory
Linear codes
Mathematical analysis
Mathematics
Matrix algebra
Matrix methods
Multiplication
Original Paper
Permutations
Rings (mathematics)
Symbolic and Algebraic Manipulation
Theory of Computation
Codes
Additive
Mass formulas
Type IV codes
Secondary 16 A10
Rings
Primary 94 B05
Type IV codes Mathematics Subject Classification Primary 94 B05
Additive 4 -codes
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Summary:There is a local ring I of order 4,  without identity for the multiplication, defined by generators and relations as I = ⟨ a , b ∣ 2 a = 2 b = 0 , a 2 = b , a b = 0 ⟩ . We give a natural map between linear codes over I and additive codes over 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 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 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.
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ISSN:0938-1279
1432-0622
DOI:10.1007/s00200-021-00488-6