Quasi Type IV codes over a non-unital ring

Adel Alahmadi (1) , Alaa Altassan , Widyan Basaffar , Alexis Bonnecaze (2) , Hatoon Shoaib (3) , Patrick Solé (2)

1 MECAA - Middle East Center of Algebra and its Applications
2 I2M - Institut de Mathématiques de Marseille
3 King Abdulaziz University

There is a local ring I of order 4, without identity for the multiplication , defined by generators and relations as I = a, b | 2a = 2b = 0, a 2 = b, ab = 0. We study the algebraic structure of linear codes over this non-unital local ring, in particular their residue and torsion codes. 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. 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 quasi Type IV codes, and classify both of them in short lengths.

Mots clés: additive F 4 -codes; mass formulas; Type IV codes MSC(2010): Primary 94 B05; Secondary 16 A10 *; rings; codes

Domaines: Mathématiques [math]

Adel Alahmadi, Alaa Altassan, Widyan Basaffar, Alexis Bonnecaze, Hatoon Shoaib, et al.. Quasi Type IV codes over a non-unital ring. Applicable Algebra in Engineering, Communication and Computing, 2021. ⟨hal-02544399⟩