Equivalence Of Codes Research Articles

Two approaches to the extension problem for arbitrary weights over finite module alphabets

The extension problem underlies notions of code equivalence. Two approaches to the extension problem are described. One is a matrix approach that reduces the general problem for weights to one for symmetrized weight compositions. The other is a monoid algebra approach that reframes the extension problem in terms of modules over the monoid algebra determined by the multiplicative monoid of a finite ring.

Equivalence of codes for countable sets of reals

AbstractA set $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ is universal for countable subsets of ${\mathbb {R}}$ if and only if for all $x \in {\mathbb {R}}$ , the section $U_x = \{y \in {\mathbb {R}} : U(x,y)\}$ is countable and for all countable sets $A \subseteq {\mathbb {R}}$ , there is an $x \in {\mathbb {R}}$ so that $U_x = A$ . Define the equivalence relation $E_U$ on ${\mathbb {R}}$ by $x_0 \ E_U \ x_1$ if and only if $U_{x_0} = U_{x_1}$ , which is the equivalence of codes for countable sets of reals according to U. The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $E_{U^*}$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $E_U$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $E_U$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $\mathbf {\Sigma }_3^1$ -generic absoluteness, then for all $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ that are $\mathbf {\Sigma }_1^1$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $E_U$ .

Bounds and Constructions of Locally Repairable Codes: Parity-Check Matrix Approach

A locally repairable code (LRC) is a linear code such that every code symbol can be recovered by accessing a small number of other code symbols. In this paper, we study bounds and constructions of LRCs from the viewpoint of parity-check matrices. Firstly, a simple and unified framework based on parity-check matrix to analyze the bounds of LRCs is proposed, and several new explicit bounds on the minimum distance of LRCs in terms of the field size are presented. In particular, we give an alternate proof of the Singleton-like bound for LRCs first proved by Gopalan et al. Some structural properties on optimal LRCs that achieve the Singleton-like bound are given. Then, we focus on constructions of optimal LRCs over the binary field. It is proved that there are only five classes of possible parameters with which optimal binary LRCs exist. Moreover, by employing the proposed parity-check matrix approach, we completely enumerate all these five classes of optimal binary LRCs attaining the Singleton-like bound in the sense of equivalence of linear codes.

Equivalence and characterizations of linear rank-metric codes based on invariants

We show that the sequence of dimensions of the linear spaces, generated by a given rank-metric code together with itself under several applications of a field automorphism, is an invariant for the whole equivalence class of the code. The same property is proven for the sequence of dimensions of the intersections of itself under several applications of a field automorphism. These invariants give rise to easily computable criteria to check if two codes are inequivalent. We derive some concrete values and bounds for these dimension sequences for some known families of rank-metric codes, namely Gabidulin and (generalized) twisted Gabidulin codes. We then derive conditions on the length of the codes with respect to the field extension degree, such that codes from different families cannot be equivalent. Furthermore, we derive upper and lower bounds on the number of equivalence classes of Gabidulin codes and twisted Gabidulin codes, improving a result of Schmidt and Zhou for a wider range of parameters. In the end we use the aforementioned sequences to determine a characterization result for Gabidulin codes.

Verification supported refactoring of embedded sql

Improving code quality without changing its functionality, e.g., by refactoring or optimization, is an everyday programming activity. Good programming practice requires that each such change should be followed by a check if the change really preserves the code behavior. If such a check is performed by testing, it can be time consuming and still cannot guarantee the absence of differences in behavior between two versions of the code. Hence, tools that could automatically verify code equivalence would be of great help. An area that we are focused on is embedded sql programming. There are a number of approaches for dealing with equivalence of either pairs of imperative code fragments or pairs of sql statements. However, in database-driven applications, simultaneous changes (changes that include both sql and a host language code) are also present and important. Such changes can preserve the overall equivalence without preserving equivalence of these two parts considered separately. In this paper, we propose an automated approach for dealing with equivalence of programs after such changes, a problem that is hardly tackled in literature. Our approach uses our custom first-order logic modeling of sql queries that corresponds to imperative semantics. The approach generates equivalence conditions that can be efficiently checked using smt solvers or first-order logic provers. We implemented the proposed approach as a framework sqlav, which is publicly available and open source.

Development and content validity of the Abilitator: a self-report questionnaire on work ability and functioning aimed at the population in a weak labour market position

BackgroundThe unemployed have lower work ability and poorer health than the employed. This situation deteriorates when unemployment continues. The long-term unemployed often have co-morbidities and face many other challenges. This increases the need for a multidimensional assessment of work ability and functioning in different service settings. In this study, we describe the development and analyse the content validity of the Abilitator, a self-report questionnaire on work ability and functioning for those in a weak labour market position.MethodsThe Abilitator was developed in 2014–2017. Its construct was assessed by members of academic expert panels (n = 30), practical expert panels of professionals (n = 700) and target group clients (n = 28). The structure and the content of the questionnaire was co-developed in 29 workshops and adjusted twice based on the expert panels’ feedback. The Abilitator was also implemented among target group clients (n = 3360) in different services and projects. During its development the Abilitator was linked to the International Classification of Functioning, Disability and Health (ICF). The content validation process followed the guidelines recommended by the Consensus-based Standards for the selection of health Measurement Instruments (COSMIN) panel.ResultsThe construct of the Abilitator combines the multidimensional and biopsychosocial models of work ability and functioning. It also includes aspects of social inclusion and employability. It evaluates social, psychological, cognitive and physical functioning, and the ability to cope with everyday life. The content of these concepts was validated by the academic and practical expert panels. The Abilitator’s 79 ICF codes covered 57% of the Generic, 77% of the Brief Vocational Rehabilitation, and 8% of the Minimal Environmental ICF Core Sets. When compared with the Work Ability Index (WAI) and the World Health Organization Disability Assessment Schedule (WHODAS 2.0), the direct equivalences of the ICF codes were 36 and 44%, respectively.ConclusionThe Abilitator sufficiently comprehensively covers the relevant aspects to enable the assessment of the overall work ability and functioning of the population in a weak labour market position.

Some generalizations of the ASR search algorithm for quasitwisted codes

One of the most important and challenging problems in coding theory is explicit construction of linear codes with the best possible parameters. It is well known that the class of quasitwisted (QT) codes is asymptotically good and contains many linear codes with best known parameters (BKLCs). A search algorithm (ASR) on QT codes has been particularly effective to construct such codes. Recently, the ASR algorithm was generalized based on the notion of code equivalence. In this work, we introduce a new generalization of the ASR algorithm to include a broader scope of QT codes. As a result of implementing this algorithm, we have found eight new linear codes over the field 𝔽 5 . Furthermore, we have found seven additional new codes from the standard constructions of puncturing, shortening or Construction X. We also introduce a new search algorithm that can be viewed as a further generalization of ASR into the class multitwisted (MT) codes. Using this method, we have found many codes with best known parameters with more direct and desirable constructions than what is currently available in the database of BKLCs.

On equivalence of maximum additive symmetric rank-distance codes

Let $${\mathcal {S}}_m({\mathbb {F}})$$ denote the set of symmetric matrices over a finite field $${\mathbb {F}}$$. Let $${\mathcal {C}}$$ be an additive subset of $${\mathcal {S}}_m({\mathbb {F}})$$ which has minimum rank-distance d and $$|{\mathcal {C}}|$$ meets the upper bound. We call $${\mathcal {C}}$$ a maximum additive d-code. In particular, when $$d=m$$, $${\mathcal {C}}$$ is equivalent to a symplectic semifield spreadset. For a given invertible matrix P, $$S_0\in {\mathcal {S}}_m({\mathbb {F}})$$ and nonzero $$a\in {\mathbb {F}}$$, it is easy to see that the map $$M \mapsto aP^T M^\sigma P + S_0$$ defines an isometry on $${\mathcal {S}}_m({\mathbb {F}})$$ with respect to the rank-metric. However, by examples of symplectic semifields, it is already known that there exist other maps preserving the rank-distance of maximum additive m-codes. In this paper, we prove that this can never happen when $$d<m$$. We also present a new construction of maximum additive 2-codes in $${\mathcal {S}}_m({\mathbb {F}})$$ for odd m which are not t-designs.

A Novel ASCII Code-based Polybius Square Alphabet Sequencer as Enhanced Cryptographic Cipher for Cyber Security Protection (APSAlpS-3CS)

For all industries, cybersecurity is regarded as one of the major areas of concern that needs to be addressed. Data and information in all forms should be safeguarded to avoid leakage of information, data theft, and robbery through intruders and hackers. This paper proposes a modification on the traditional 5x5 Polybius square in cryptography, through dynamically generated matrices. The modification is done through shifting cell elements for every encrypted character using a secret key and its ASCII decimal code equivalents. The results of the study revealed that the modified Polybius cipher offers a more secure plaintext-ciphertext conversion and is difficult to break, as evident in the frequency analysis. In the proposed method, each element produced in the digraphs exhibits a wider range of possible values. However, with the increase of process in the encryption and decryption, the modified Polybius cipher obtained a longer execution time of 0.0031ms, being identified as its tradeoff. The unmodified Polybius cipher, however, obtained an execution time of 0.0005ms. Future researchers may address the execution time tradeoff of the modified Polybius cipher.

Local equivalence of qudit color codes and toric codes

In this paper, we study nonbinary toric codes and color codes. While the relationships between the qubit color codes and toric codes have been studied extensively from various perspectives, similar studies for qudit toric codes and color codes are missing. We show that just as in the binary case, qudit color codes can be mapped to qudit toric codes. Qudit color codes are known to have the capability for implementing the generalized Clifford group transversally. The proposed mapping of qudit color codes to toric codes would allow us to port the computational capabilities of the qudit color codes to qudit toric codes. Another application would be to decode qudit color codes via qudit toric codes. Along the way, we show an important structural result for higher-dimensional qudit toric codes. Specifically, we show that the toric codes over prime power dimension can be viewed as a direct sum of copies of codes over the prime dimension. This result simplifies the study of codes over prime power dimension by reducing them to the study of codes over the prime dimension. Furthermore, we provide explicit circuits for the transformation between qudit color codes and toric codes.

Linear complementary dual codes over rings

By using linear algebra over finite commutative rings, we will present some judging criterions for linear complementary dual (LCD) codes over rings, in particular, free LCD codes over finite commutative rings are described. By using free LCD codes over finite commutative rings and the Chinese Remainder Theorem, LCD codes over semi-simple rings are constructed and the equivalence of free codes and free LCD codes is given. In addition, all the possible LCD codes over chain rings are determined. We also generalize the judging criterion for cyclic LCD codes over finite fields to cyclic LCD codes over chain rings. Based on the above results and the Chinese Remainder Theorem, we also present results for LCD codes over principal ideal rings.

Adaptation and equivalence of codes in the design of a board game for the visually impaired: graphic and haptic code

La majoria de comunicacions grafiques requereixen l'organ de la visio. Es per aixo que son inaccessibles per a les persones amb discapacitat visual. En aquesta investigacio s'analitzen els parametres de disseny de jocs de taula classics, s'identifiquen aquells elements que influeixen en el desenvolupament del joc i es proposa una equivalencia de codis. El metode es va iniciar amb la definicio de les variables independents (textura i grandaria) i la dependent (identificacio d'elements), es van construir els prototips d'un joc concret i es va verificar la proposta mitjancant un test de recepcio. Dels resultats, s'estableix l'equivalencia de codis perque els jocs puguin ser usats tant per persones amb discapacitat visual com per les que no la tenen, complint amb el disseny inclusiu.

Identifiers for MRD-codes

For any admissible value of the parameters n and k there exist [n,k]-Maximum Rank Distance Fq-linear codes. Indeed, it can be shown that if field extensions large enough are considered, almost all rank metric codes are MRD. On the other hand, very few families up to equivalence of such codes are currently known. In the present paper we study some invariants of MRD codes and evaluate their value for the known families, providing a new characterization of generalized twisted Gabidulin codes.

On equivalence of cyclic codes, generalization of a quasi-twisted search algorithm, and new linear codes

A fundamental problem in coding theory is the explicit construction of linear codes with best possible parameters. A search algorithm (ASR) on certain types of quasi-twisted (QT) codes has been very fruitful to address this challenging problem. In this work, we generalize the ASR algorithm to make it more comprehensive. The generalization is based on code equivalence. As a result of implementing the more general algorithm, we discovered 27 new linear codes over the fields $$\mathbb {F}_q$$ for $$q=3,4,5,$$ and 7. Further, we prove several useful theoretical results about the equivalence of cyclic codes, constacyclic codes, and QT codes.

Relative Generalized Matrix Weights of Matrix Codes for Universal Security on Wire-Tap Networks

Universal security over a network with linear network coding has been intensively studied. However, previous linear codes and code pairs used for this purpose were linear over a larger field than that used on the network, which restricts the possible packet lengths of optimal universal secure codes, does not allow to apply known list-decodable rank-metric codes and requires performing operations over a large field. In this paper, we introduce new parameters (relative generalized matrix weights and relative dimension/rank support profile) for code pairs that are linear over the field used in the network, and show that they measure the universal security performance of these code pairs. For one code and non-square matrices, generalized matrix weights coincide with the existing Delsarte generalized weights, hence we prove the connection between these latter weights and secure network coding, which was left open. As main applications, the proposed new parameters enable us to: 1) obtain optimal universal secure linear codes on noiseless networks for all possible packet lengths, in particular for packet lengths not considered before, 2) obtain the first universal secure list-decodable rank-metric code pairs with polynomial-sized lists, based on a recent construction by Guruswami et al; and 3) obtain new characterizations of security equivalences of linear codes. Finally, we show that our parameters extend relative generalized Hamming weights and relative dimension/length profile, respectively, and relative generalized rank weights and relative dimension/intersection profile, respectively.

On the equivalence of cyclic and quasi-cyclic codes over finite fields

This paper studies the equivalence problem for cyclic codes of length $p^r$ and quasi-cyclic codes of length $p^rl$. In particular, we generalize the results of Huffman, Job, and Pless (J. Combin. Theory. A, 62, 183--215, 1993), who considered the special case $p^2$. This is achieved by explicitly giving the permutations by which two cyclic codes of prime power length are equivalent. This allows us to obtain an algorithm which solves the problem of equivalency for cyclic codes of length $p^r$ in polynomial time. Further, we characterize the set by which two quasi-cyclic codes of length $p^rl$ can be equivalent, and prove that the affine group is one of its subsets.

Квадрат кода Рида - Маллера и классы эквивалентности секретных ключей криптосистемы Мак-Элиса - Сидельникова

Квадрат кода Рида - Маллера и классы эквивалентности секретных ключей криптосистемы Мак-Элиса - Сидельникова

Codes over an infinite family of algebras

In this paper, we will show some properties of codes over the ring $B_k=\mathbb{F}_p[v_1,\dots,v_k]/(v_i^2=v_i,\forall i=1,\dots,k).$ These rings, form a family of commutative algebras over finite field $\mathbb{F}_p$. We first discuss about the form of maximal ideals and characterization of automorphisms for the ring $B_k$. Then, we define certain Gray map which can be used to give a connection between codes over $B_k$ and codes over $\mathbb{F}_p$. Using the previous connection, we give a characterization for equivalence of codes over $B_k$ and Euclidean self-dual codes. Furthermore, we give generators for invariant ring of Euclidean self-dual codes over $B_k$ through MacWilliams relation of Hamming weight enumerator for such codes.

Algorithms for Group Isomorphism via Group Extensions and Cohomology

The isomorphism problem for finite groups of order $n$ (GpI) has long been known to be solvable in $n^{\log n+O(1)}$ time, but only recently were polynomial-time algorithms designed for several interesting group classes. Inspired by recent progress, we revisit the strategy for GpI via the extension theory of groups. The extension theory describes how a normal subgroup $N$ is related to $G/N$ via $G$, and this naturally leads to a divide-and-conquer strategy that “splits” GpI into two subproblems: one regarding group actions on other groups, and one regarding group cohomology. When the normal subgroup $N$ is abelian, this strategy is well known. Our first contribution is to extend this strategy to handle the case when $N$ is not necessarily abelian. This allows us to provide a unified explanation of all recent polynomial-time algorithms for special group classes. Guided by this strategy, to make further progress on GpI, we consider central-radical groups, proposed in Babai et al. [Code equivalence and group isomorphism, in Proceedings of the 22nd Annual ACM--SIAM Symposium on Discrete Algorithms (SODA'11), SIAM, Philadelphia, 2011, ACM, New York, pp. 1395--1408]: the class of groups such that $G$ modulo its center has no abelian normal subgroups. This class is a natural extension of the group class considered by Babai et al. [Polynomial-time isomorphism test for groups with no abelian normal subgroups (extended abstract), in International Colloquium on Automata, Languages, and Programming (ICALP), 2012, pp. 51--62], namely those groups with no abelian normal subgroups. Following the above strategy, we solve GpI in $n^{O(\log \log n)}$ time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GpI in $n^{O(\log\log n)}$ time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time there have been worst-case guarantees on an $n^{o(\log n)}$-time algorithm that tackles both aspects of GpI---actions and cohomology---simultaneously. Prior to this work, the best proven upper bounds on algorithms for groups with central radicals were $n^{O(\log n)}$, even for groups with a central radical of constant size, such as ${Rad}(G) = Z(G)=\mathbb{Z}_2$. To develop our new algorithms we utilize several mathematical results on the detailed structure of cohomology classes, as well as algorithmic results for code equivalence, coset intersection, and cyclicity testing of modules over finite-dimensional associative algebras. We also suggest several promising directions for future work.

Rimini protokoll: Adolf Hitler: Mein kampf, vol. 1 & 2: Il Teatro di fronte alla Storia

Within the sphere of contemporary performative theatre – where the text cedes its leading role as bearer of the author’s message, in favour of the equivalence of various expressive codes – lies the work of Rimini Protokoll: a German team of author-directors that has been active in Europe for more than fifteen years with traditional and interactive theatre, installations, sound and radio plays. Rimini Protokoll’s work is also an expression of German documentary theatre, which reflects on the problematic nature of history by focusing on unresolved issues in the present. In this framework, this article proposes an analysis of Adolf Hitler: Mein Kampf, Vol. 1 & 2, an example of a performance that is centred on a burning contemporary topic: the end of the copyright on Hitler’s Mein Kampf and the subsequent problem of its reprint. On the stage there are six “experts” – not professional actors – who examine their personal relationship with the book and most of all investigate its influence on the present. This starting-point is an opportunity for a wider, more articulated reflection on social and historical issues that go beyond Germany in the Nazi period, but spread across time and space. The urgency is not the reconstruction of the past, but its repercussion on the present. The performance is furthermore a reflection on the role of books in the collective imagination and on their connection with other media.