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Errata for "The linear programming bound for codes over finite Frobenius rings"

Eimear Byrne, Marcus Greferath, Michael E. O'Sullivan
2007 Designs, Codes and Cryptography  
The validity of its results is not affected, nor is that of the examples since their computation did not involve the dual version of the LP bound.  ...  The authors discovered some mistakes in the article that appeared on pp. 289-301 of the March volume of DCC 42 (2007) .  ...  The dual version of the linear programming bound In Sect. 3, R is a finite Frobenius ring and θ and τ are equivalence relations on R.  ... 
doi:10.1007/s10623-007-9120-3 fatcat:tpw3hzfgwnfgxgiri5h24jkgee

New Bounds for Codes Over Finite Frobenius Rings [article]

Eimear Byrne, Marcus Greferath, Axel Kohnert, Vitaly Skachek
2009 arXiv   pre-print
We give results on the question of code optimality for linear codes over finite Frobenius rings for the homogeneous weight.  ...  This article improves on the existing Plotkin bound derived in an earlier paper, and suggests a version of a Singleton bound. We also present some families of codes meeting these new bounds.  ...  In this note we present further bounds for linear codes over finite Frobenius rings for the homogeneous weight. We give a refinement of the Plotkin bound given in [6] .  ... 
arXiv:0905.1313v1 fatcat:pd2rzrj3bvbzfhf32vm65do3wm

Quaternions over Galois rings and their codes [article]

Pierre Lance Tan, Virgilio Sison
2021 arXiv   pre-print
Finally, one-sided linear block codes over the quaternions over Galois rings are constructed, and certain bounds on the homogeneous distance of the images of these codes are proved.  ...  It is shown in this paper that, if R is a Frobenius ring, then the quaternion ring ℋ_a,b(R) is a Frobenius ring for all units a,b ∈ R.  ...  Acknowledgement The authors gratefully acknowledge the members of the IMSP Coding Theory and Cryptography Research Cluster for helpful discussions.  ... 
arXiv:2109.00735v1 fatcat:ut7qe4ycljeslb4n5uxj6nfmmq

Page 5701 of Mathematical Reviews Vol. , Issue 2003g [page]

2003 Mathematical Reviews  
The author proves an extension theorem for linear codes over a finite quasi-Frobenius module, namely: Let rgQ be a quasi- Frobenius module over a finite commutative ring R with | and K Cr Q be a linear  ...  In many situations, Delsarte’s linear programming bound [P. Del- sarte, Philips Res. Rep. Suppl.  ... 

Do non-free LCD codes over finite commutative Frobenius rings exist? [article]

Sanjit Bhowmick and Alexandre Fotue-Tabue and Edgar Martínez-Moro and Ramakrishna Bandi and Satya Bagchi
2019 arXiv   pre-print
We then obtain a necessary and sufficient condition for the existence of LCD code over finite commutative Frobenius rings.  ...  In this paper, we clarify some aspects on LCD codes in the literature. We first prove that a non-free LCD code does not exist over finite commutative Frobenius local rings.  ...  Acknowledgements The first author of the paper would like to Ministry of Human Resource and Development India for support financially to carry out this work.  ... 
arXiv:1901.10836v1 fatcat:ppmlrsu7nndd5ie5jjrlhlpa4u

Entanglement-Assisted Quantum Error-Correcting Codes over Local Frobenius Rings [article]

Tania Sidana, Navin Kashyap
2022 arXiv   pre-print
We derive a formula for the minimum number of entanglement qudits required to construct an EAQECC from a linear code over the ring ℤ_p^s.  ...  In this paper, we provide a framework for constructing entanglement-assisted quantum error-correcting codes (EAQECCs) from classical additive codes over a finite commutative local Frobenius ring.  ...  Constructing EAQECCs from Codes over Local Frobenius Rings In this section, we provide the details of the method for constructing EAQECCs from additive codes over a finite commutative local Frobenius ring  ... 
arXiv:2202.00248v3 fatcat:lc6bulfmrzflrfjiqw5h73kvb4

Coding Theory (Dagstuhl Seminar 11461)

Joachim Rosenthal, M. Armin Shokrollahi, Judy L. Walker, Marc Herbstritt
2012 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 11461 "Coding Theory". A (channel) code is typically a set of vectors of the same length n over a finite alphabet Σ.  ...  Coding theory is all about the design of excellent codes as a function of the communications channel, and the design of efficient algorithms for choosing the codebook vectors, and more importantly, for  ...  As a by-product we obtain a very nice new characterization of the class of all Frobenius rings. This is given as follows: Theorem: For a finite ring R the following are equivalent.  ... 
doi:10.4230/dagrep.1.11.50 dblp:journals/dagstuhl-reports/RosenthalSW11 fatcat:vsvcwodxnzc4rpd2lxi4w6zkae

Page 1711 of Mathematical Reviews Vol. , Issue 2000c [page]

2000 Mathematical Reviews  
Given q and g, let N,(g) be the maximal number of rational points on curves of genus g over F,. Usually, Oesterlé’s linear programming method is used to obtain the upper bound on N,(g).  ...  The same idea is used in this paper to show that the exact value of N3(5) is smaller than the upper bound from Oéesterlé’s linear programming method.  ... 

Page 1629 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
We introduce a linear programming problem for bound- ing these linear forms from below.  ...  Summary: “In this paper we completely describe the 2n + 2 min- imal cyclic codes of length 2p” over F, as minimal ideals in the ring R =F,[x]/(x??" — 1) in terms of their generating idempo- tents.  ... 

The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices [article]

Steven T. Dougherty, Jon-Lark Kim, Buket Ozkaya, Lin Sok, Patrick Solé
2015 arXiv   pre-print
We give a linear programming bound on the largest size of an LCD code of given length and minimum distance.  ...  We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings R_k.  ...  The orthogonal is defined by C ⊥ = {v ∈ R n | [v, w] = 0, ∀w ∈ C}. If R is a Frobenius ring then we have that |C||C ⊥ | = |R| n . For codes over finite fields we have dim(C) + dim(C ⊥ ) = n.  ... 
arXiv:1506.01955v1 fatcat:hpku65anlfcwlcv7db4u4lwn6m

Quantum Cyclic Code of length dividing p^t+1 [article]

Sagarmoy Dutta, Piyush P Kurur
2011 arXiv   pre-print
In this paper, we study cyclic stabiliser codes over F_p of length dividing p^t+1 for some positive integer t. We call these t-Frobenius codes or just Frobenius codes for short.  ...  An important subclass of stabiliser codes are the linear stabiliser codes. For linear Frobenius codes we have stronger results: We completely characterise all linear Frobenius codes.  ...  Moreover, this impossibility is purely Galois theoretic unlike other known restriction that arise from sphere packing bounds or linear programming bounds.  ... 
arXiv:1011.5814v2 fatcat:akg7l5rrh5bs3asnal7xt6ozba

Cyclic codes over a non-chain ring R_e,q and their application to LCD codes [article]

Habibul Islam and Edgar Martínez-Moro and Om Prakash
2021 arXiv   pre-print
Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over R_e,q.  ...  In this paper, we study cyclic codes of length n over a non-chain ring R_e,q=𝔽_q[u]/⟨ u^e-1⟩.  ...  Also, authors would like to thank the anonymous referee(s) and the Editor for their valuable comments to improve the presentation of the manuscript.  ... 
arXiv:2106.07962v1 fatcat:lyxkkyvbt5de5d77wrtizw2swi

On Langlands program, related representation and G-shtukas [article]

Nikolaj Glazunov
2020 arXiv   pre-print
This communication is an introduction to the Langlands Program and to (G-) shtukas (over algebraic curves) over function fields.  ...  For a connected reductive group G over a global field K, the Langlands correspondence relates automorphic forms on G and global Langlands parameters, i.e. conjugacy classes of homomorphisms from the Galois  ...  At first recall that linear algebraic groups found important applications in the Langlands program.  ... 
arXiv:2004.11415v1 fatcat:p7vj5hx3mzbrhhzhzlb2fqtahi

Page 3040 of Mathematical Reviews Vol. , Issue 89F [page]

1989 Mathematical Reviews  
The second chapter, on codes and designs, has two principal themes: (i) the links between codes that have optimal parameters, and block designs; and (ii) t-designs that live in linear codes.  ...  Regarding (i), the well-known connection between the Plotkin bounds and Hadamard 2-designs is explored; analogous results on equidistant codes and constant weight codes, based mostly on the work of N.  ... 

Page 6018 of Mathematical Reviews Vol. , Issue 2001H [page]

2001 Mathematical Reviews  
The authors develop a general theory of linear codes over finite rings and modules. The notions of systematic code, dual code, parity check matrix and Hamming distance are introduced.  ...  (RS-MOSC-NIT; Moscow) Linear codes over finite rings and modules. (Russian. English and Russian summaries) Functional analysis, differential equations and their applications (Russian) (Puebla, 1995).  ... 
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