[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83637-en":3,"doc-seo-83637-105":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},83637,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Generalized Extended Codes with Applications in Entanglement-Assisted Qubit and Qutrit Codes","Generalized extended codes are constructed from linear codes over F_{q^2} using an extension vector u and scalar a, extending the earlier second-kind construction and encompassing all linear codes [n,k,d]_{q^2} with d≥2 up to permutation equivalence. Their Hermitian dual structure is analyzed via monomial equivalence, with explicit criteria controlling the Hermitian hull and the Hermitian dual distance through the relative position of u and interactions with minimum-weight codewords of C^{⊥_H}. Applying these criteria yields 267 new EA qubit codes (n≤40) and 14 new EA qutrit codes (n≤25) with proven parameter improvements over known tables.","arXiv :2607 .02 170v 1 [ cs .IT] 2 Jul 2026  \nGeneralized Extended Codes with Applications in Entanglement-Assisted Qubit and Qutrit Codes  \nYang Li, Martianus Frederic Ezerman, Shitao Li, San Ling, and Zhonghua Sun  \nAbstract  \nGiven a linear code C of length n over Fq2 and a nonzero vector u ∈ Fnq2 , Sun, Ding, and Chen introduced the second kind of extended construction resulting in the code C (u) . Their construction generalizes the standard extended construction C(−1), that is, when u is fixed to −1 . They further showed that every [n, k, d]q2 linear code with d ≥ 2 can be obtained from this construction for a suitable choice of the initial code C and the extension vector u, up to permutation equivalence. To construct entanglement-assisted quantum error-correcting codes (EAQECCs) with more flexible and potentially better parameters, we consider the Hermitian dual of such extended linear codes and apply monomial variations on them. The resulting family of linear codes consist of q2-ary codes C with respect to vector u ∈ Fnq2 \\ C and the scalar a ∈ F∗q2 . We call them generalized extended codes, with the code denoted by C (u, a) .  \nWe prove that any generalized extended code is monomially equivalent to the Hermitian dual of a code which is closely related to a second kind of extended code of C ⊥ H . Every [n + 1, k + 1]q2 linear code D with d(D⊥ H) > 1 is monomially equivalent to the generalized extended code C (u, a) of an [n, k]q2 linear code C for a fixed a ∈ F∗q2 and some u ∈ Fnq2 . We then characterize the Hermitian hull and Hermitian dual distance of C(u, a) in terms of the position of u relative to C + C⊥ H and the interaction between u and the minimum weight codewords of C ⊥ H, respectively. We obtain explicit criteria to independently control the expected Hermitian hull dimension and Hermitian dual distance of C(u, a) . In particular, several conditions for simultaneously increasing the Hermitian hull dimension and the Hermitian dual distance of C(u, a) are derived.  \nApplying these results to the Hermitian construction for EAQECCs gives us 267 new EA qubit codes of lengths n ≤ 40 and  \n14 new EA qutrit codes of lengths n ≤ 25 compared to the best-known codes in Grassl’s code tables and the imporvements recorded in very recent works in the literature. Among the new parameter sets, we confirm improvements for 236 qubit and 8 qutrit codes.  \nIndex Terms  \nEntanglement-assisted, quantum code, Hermitian dual distance, Hermitian hull, generalized extended code.  \nI. INTRODUCTION  \nTHROUGHOUT this paper, let Fq denote the finite field with q elements and let F∗q = Fq \\ {0}, where q is a prime power.  \nAn [n, k, d]q2 linear code C is a k-dimensional linear subspace of Fnq2 with minimum Hamming distance d := d(C) . We use 0 and 1 to denote appropriate all zero and all one vectors, respectively.  \nA. Hermitian Hulls and Equivalent Linear Codes  \nThe Hermitian inner product of any two vectors x = (x1 , x2 ,..., xn) and y = (y1 , y2 ,..., yn) in Fnq2 is ⟨x, y⟩H = P xi yqi . The Hermitian dual and the Hermitian hull of C are defined, respectively, by  \nC ⊥H = {y ∈ Fnq2 : ⟨x, y⟩H = 0 for all x ∈ C} and HullH (C) = C ∩ C⊥H .  \nThe code C is Hermitian self-orthogonal if HullH (C) = C. The code is Hermitian dual-containing if HullH (C) = C ⊥H . It is Hermitian self-orthogonal if and only if C ⊥H is Hermitian dual-containing. If C is an [n, k]q2 linear code, then C ⊥H is an [n, n − k]q2 linear code and HullH (C) is an [n,ℓ]q2 Hermitian self-orthogonal code, with ℓ being the hull dimension.  \nA linear code C can be completely described by a generator matrix G := G(C) whose rows form a basis. A generator matrix H of C ⊥H is a Hermitian parity-check matrix of C. Two linear codes C1 and C2 that can be generated, respectively, by G 1 and G2 are permutation equivalent if there exists a permutation matrix P such that G 1 P is a generator matrix of C2 . The two codes are monomially equivalent if there exists a monomial matrix M such that ","cbCaihKAF82QP9z4","https://ap.wps.com/l/cbCaihKAF82QP9z4","pdf",644346,3,1,23,"English","en",105,"# Abstract\n## Hermitian hulls and equivalent linear codes\n## EAQECCs from the second kind of extended codes","[{\"question\":\"How are generalized extended codes defined in the paper?\",\"answer\":\"They are built from a linear code C over F_{q^2} using an extension vector u and a scalar a, denoted by C(u,a), with u chosen relative to C (and C^{⊥_H}) and a from F^*_{q^2}.\"},{\"question\":\"What role does monomial equivalence play in the results?\",\"answer\":\"The paper proves that any generalized extended code is monomially equivalent to the Hermitian dual of a code closely related to a second-kind extended code of C^{⊥_H}, enabling a structural analysis of parameters.\"},{\"question\":\"How are the Hermitian hull dimension and Hermitian dual distance controlled?\",\"answer\":\"The criteria express these quantities in terms of where u lies with respect to C + C^{⊥_H} and how u interacts with minimum-weight codewords of C^{⊥_H}, allowing independent or simultaneous increase under certain conditions.\"}]",1784189423,58,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"generalized-extended-codes-with-applications-in-entanglement-assisted-qubit-and-qutrit-codes","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/generalized-extended-codes-with-applications-in-entanglement-assisted-qubit-and-qutrit-codes/83637/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How are generalized extended codes defined in the paper?","Question",{"text":75,"@type":76},"They are built from a linear code C over F_{q^2} using an extension vector u and a scalar a, denoted by C(u,a), with u chosen relative to C (and C^{⊥_H}) and a from F^*_{q^2}.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What role does monomial equivalence play in the results?",{"text":80,"@type":76},"The paper proves that any generalized extended code is monomially equivalent to the Hermitian dual of a code closely related to a second-kind extended code of C^{⊥_H}, enabling a structural analysis of parameters.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the Hermitian hull dimension and Hermitian dual distance controlled?",{"text":84,"@type":76},"The criteria express these quantities in terms of where u lies with respect to C + C^{⊥_H} and how u interacts with minimum-weight codewords of C^{⊥_H}, allowing independent or simultaneous increase under certain 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