[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86515-en":3,"doc-seo-86515-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86515,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","Construction of Self-Orthogonal Codes over a Commutative Non-Unitary Ring of Order 25","Codes over non-unitary rings are examined, focusing on the commutative non-unitary ring Ip of Fine classification with order p2. For p=2 and p=3, three families—self-orthogonal, quasi self-dual, and self-dual codes over Ip—have been studied using mass formulas and building-up constructions up to equivalence for small lengths. This work treats p=5, introducing linear I5-codes, their three categories, and relations to residue and torsion codes, with complete classification in lengths ≤4 and corrections to prior mass-formula-based results.","arXiv :2607 . 10844v 1 [ cs .IT] 12 Jul 2026  \nConstruction of self-orthogonal codes over a commutative non-unitary ring of order 25 ∗†  \nJon-Lark Kim  \nDepartment of Mathematics and Institute for Mathematical and Data Sciences  \nSogang University, Seoul, Korea [jlkim@sogang. ac. kr](jlkim@sogang. ac. kr)  \nMarvin Olavides‡  \nDepartment of Mathematics and Institute for Mathematical and Data Sciences Sogang University, Seoul, Korea [mmolavides@gmail. com](mmolavides@gmail. com)  \nYoung Gun Roe  \nKNU Research Institute for Mathematical Sciences Kangwon National University, Chuncheon, Korea [ygroe@naver. com](ygroe@naver. com)  \nAbstract  \nCodes over non-unitary rings have been studied recently. In particular, codes over the commutative non-unitary ring Ip (in the classification of Fine) of order p2 where p is a prime are being considered. For p “ 2 (resp. p “ 3), three categories of codes over Ip have been studied: self-orthogonal codes, quasi self-dual codes, and self-dual codes over Ip. Using some related mass formulas and building-up constructions, classifications of  \n∗ J.-L. Kim was supported in part by the BK21 FOUR (Fostering Outstanding Universities for Research) funded by the Ministry of Education (MOE, Korea) and National Research Foundation of Korea (NRF) under Grant No. 4120240415042 and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT under Grant No. RS-2025-24534992 .  \n†Y.G. Roe was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (RS-2025-25415913) .  \n‡Corresponding author  \nthese codes have been done up to the permutation equivalence (resp. the monomial equivalence) for certain small lengths. In this paper, we take the prime p “ 5 and consider the ring I5 . We introduce the notion of linear codes over I5 . We also define the same three categories of linear I5-codes, study the structures of these I5-codes and relate them to their associated residue and torsion codes. We classify the three categories of codes completely in lengths at most 4 up to the monomial equivalence for a given type tk1 , k2 u. Moreover, in the paper of Alahmadi et al. regarding the mass formula for self-orthogonal codes over Ip, mistakes in the classification of quasi self-dual codes over I5 had been made such as incorrect automorphism group order of some codes or inconsistency with the mass formula for self-orthogonal codes over Ip for length n “ 2 and type t1, 0u and for length n “ 3 and type t1, 1u. We correct and improve such results.  \nKeywords : self-orthogonal codes, quasi self-dual codes, building-up construction, nonunitary rings  \nMathematics Subject Classification : 94B05, 16D10  \n1 Introduction  \nCoding theory over rings has evolved into a significant area of modern algebraic coding theory, broadening classical concepts that were once confined to finite fields. The discovery by Hammons et al. [19] that nonlinear binary codes such as the Kerdock and Preparata codes can be viewed as binary images under a Gray map of linear codes over Z4 opened an entirely new direction in the study of codes on more general algebraic structures. Since then, many works have revealed deep links between ring-based codes and topics such as lattices, modular forms, and cryptography [15,16,20] .  \nWithin this framework, self-orthogonal and self-dual codes occupy a central role. They are known for their excellent distance properties and for their close relationships with combinatorial designs and lattices. Alahmadi et al. [2] mentioned that the classification of self-dual codes over unitary rings and finite fields has rested on two pillars: an algorithm to generate short length codes and a mass formula to signal the completion of the classification [10,27] . One of the construction methods is the so called building-up method. By using a recursion on generator matrices, the building-up met","cbCaibPic9eYUibG","https://ap.wps.com/l/cbCaibPic9eYUibG","pdf",438274,4,1,29,"English","en",105,"# Introduction\n## Coding theory over rings and motivation\n## Self-orthogonal/self-dual codes and mass formulas\n## Building-up constructions and propagation rules\n## Non-unitary rings in Fine’s classification\n# Linear I5-codes and code categories\n## Residue and torsion code relations\n## Classification up to equivalence and corrections to prior work","[{\"question\":\"What types of codes over the ring Ip are considered in this paper?\",\"answer\":\"The paper studies self-orthogonal, quasi self-dual, and self-dual (in the corresponding sense) codes over Ip, focusing on the case p=5 and defining linear I5-codes for these categories.\"},{\"question\":\"How are the constructions and classifications in this work performed?\",\"answer\":\"The approach uses related mass formulas and building-up constructions, then classifies the codes completely in lengths up to 4 under monomial equivalence for the specified type.\"},{\"question\":\"What issues in earlier classification results are addressed?\",\"answer\":\"For p=5, the paper corrects mistakes in the quasi self-dual code classification, including incorrect automorphism group orders and inconsistencies with mass-formula requirements for certain lengths and 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types of codes over the ring Ip are considered in this paper?","Question",{"text":75,"@type":76},"The paper studies self-orthogonal, quasi self-dual, and self-dual (in the corresponding sense) codes over Ip, focusing on the case p=5 and defining linear I5-codes for these categories.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the constructions and classifications in this work performed?",{"text":80,"@type":76},"The approach uses related mass formulas and building-up constructions, then classifies the codes completely in lengths up to 4 under monomial equivalence for the specified type.",{"name":82,"@type":73,"acceptedAnswer":83},"What issues in earlier classification results are addressed?",{"text":84,"@type":76},"For p=5, the paper corrects mistakes in the quasi self-dual code classification, including incorrect automorphism group orders and inconsistencies with mass-formula requirements for certain lengths and 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