Dual Code Research Articles

Theory of additive complementary dual codes, constructions and computations

We study complementary dual Fq-linear codes defined over an extension field Fqm, which we refer to as additive complementary dual (ACD) codes. We first consider duality of this class of codes with respect to a general inner product on Fqmn, which covers the commonly studied cases such as the trace-Euclidean and Hermitian inner products. In general this inner product is not necessarily symmetric, hence we define the notions of left/right hulls and left/right duals of an additive code. This leads to one-sided ACD codes, for which we prove a characterization result in terms of the generator matrix, extending the analogous characterization of Massey for linear complementary dual (LCD) codes. Then we focus on constructions and parameters of ACD codes with respect to trace-Euclidean, Hermitian and Galois inner products. We prove how LCD codes yield ACD codes relative to inner products considered in this work, make some observations on MDS subclass of ACD codes and present a construction of an ACD code by expanding another ACD code. We provide extensive computational results and tables on ACD codes over F4,F8 and F9. Our computations yield many MDS ACD codes and also examples of ACD codes, which have more codewords than the comparable LCD codes with the same length and minimum distance.

New classes of affine-invariant codes sandwiched between Reed–Muller codes

Reed–Muller codes have been extensively studied due to their favorable theoretical and mathematical properties. We propose some new classes of extended cyclic codes sandwiched between generalized Reed–Muller codes Rq(r−1,n) and Rq(r,n) with n=3m, m⩾1. We show that these new codes are affine-invariant, and compute their dimensions and dual codes. We also give lower bounds for their minimum distances. In addition, we explicitly determine the minimum vectors of these codes for some special cases. Our new method can be used to construct codes of length qn with odd n, which makes up for the shortcomings of the previous construction method which works only for even n. Moreover, our new extended cyclic codes have higher rate than those in Xu et al. (2023) [17].

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A description of the algorithm for optimal symbol-by-symbol reception of signal structures based on block noise-resistant codes in non-binary Galois fields is given. It is shown that the basis of this algorithm is the fast spectral transformation algorithm in the Walsh–Hadamard basis with the dimension of the Galois field. It is shown that the resulting complexity of the analyzed character-by-character reception algorithm is determined by the dimension of the dual code, which determines the prospects of its application for block noise-resistant codes with a high code rate (with low redundancy). The results of modeling a character-by-symbol reception algorithm are presented in order to study noise immunity for a number of frequency-efficient digital signals with quadrature-amplitude and amplitude-phase manipulations (with frequency coefficient efficiencies of 3, 4 and 6 bps/Hz) in combination with a parity check code. It is shown that the use of a symbol-by-symbol reception algorithm provides an energy gain of up to 1.5...3.0 dB in relation to the transmission and reception of the considered series of signals without coding.

The Hermitian Dual Codes of Several Classes of BCH Codes

As a special subclass of cyclic codes, BCH codes are usually among the best cyclic codes and have wide applications in communication and storage systems and consumer electronics. Let <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> be a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -ary BCH code of length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> with respect to an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -th primitive root of unity β over an extension field of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>q</i>²</sub> , and let <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥<i>H</i></sup> denote its Hermitian dual code, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> is a prime power. If both <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥<i>H</i></sup> are a BCH code with respect to an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -th primitive root of unity β, then <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> is called a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Hermitian dually-BCH code</i> . The objective of this paper is to derive a necessary and sufficient condition for ensuring that two classes of narrow-sense BCH codes are Hermitian dually-BCH codes. As by-products, lower bounds on the minimum distances of the Hermitian dual codes of these BCH codes are developed, which improve the lower bounds documented in IEEE Trans. Inf. Theory, vol. 68, no. 2, pp. 953-964, 2022, in some cases.

Connections between Linear Complementary Dual Codes, Permanents and Geometry

Linear codes with complementary duals, or LCD codes, have recently been applied to side-channel and fault injection attack-resistant cryptographic countermeasures. We explain that over characteristic two fields, they exist whenever the permanent of any generator matrix is non-zero. Alternatively, in the binary case, the matroid represented by the columns of the matrix has an odd number of bases. We explain how Grassmannian varieties as well as linear and quadratic complexes are connected with LCD codes. Accessing the classification of polarities, we relate the binary LCD codes of dimension k to the two kinds of symmetric non-singular binary matrices, to certain truncated Reed–Muller codes, and to the geometric codes of planes in finite projective space via the self-orthogonal codes of dimension k.

Three classes of BCH codes and their duals

BCH codes are an important class of cyclic codes, and have wide applications in communication and storage systems. However, it is difficult to determine the parameters of BCH codes and only a few cases are known. In this paper, we mainly study three classes of BCH codes with n=qm−1,q2s−1q+1,qm−1q−1. On one hand, we provide the parameters of C(n,q,δ,1) and its dual code in some cases. On the other hand, we provide the sufficient and necessary conditions to guarantee that C(n,q,δ,2) is a dually-BCH code.

Near MDS codes of non-elliptic-curve type from Reed-Solomon codes

An [n,k,d] linear code is said to be maximum distance separable (MDS) or almost maximum distance separable (AMDS) if d=n−k+1 or d=n−k, respectively. If a code and its dual code are both AMDS, then the linear code is called a near maximum distance separable (NMDS) code. NMDS codes correspond to some interesting objects in finite geometry and have nice applications in cryptography. The NMDS codes constructed from elliptic curves are referred as elliptic curve NMDS codes. In this paper, by adding two suitable columns to the generator matrices of famous Reed-Solomon codes, we can obtain two families of NMDS codes with parameters [q+3,4,q−1]q (q is a prime power with gcd⁡(q−1,3)=1) and [2m+3,5,2m−2]2m respectively. These NMDS codes are shown to be linearly inequivalent to elliptic curve NMDS codes. Besides, the weight enumerators and locality of these NMDS codes are completely determined. It turns out that the resultant NMDS codes and their dual codes are mostly distance-optimal and dimension-optimal locally recoverable codes.

Some 3-designs and shortened codes from binary cyclic codes with three zeros

Linear codes and t-designs are interactive with each other. It is well known that some t-designs have been constructed by using certain linear codes in recent years. However, only a small number of infinite families of the extended codes of linear codes holding an infinite family of t-designs with t≥3 are reported in the literature. In this paper, we study the extended codes of the augmented codes of a class of binary cyclic codes with three zeros and their dual codes, and show that those codes hold 3-designs. Furthermore, we obtain some shortened codes from the studied cyclic codes and explicitly determine their parameters. Some of those shortened codes are optimal or almost optimal.

Several Families of Binary Minimal Linear Codes From Two-to-One Functions

Minimal linear codes have important applications in secure communications, including in the framework of secret sharing schemes and secure multi-party computation. A lot of research have been carried out to derive codes with few weights (but more importantly, being minimal) using algebraic or geometric approaches. One of the main power and fructify algebraic methods is based on the design of those codes by employing functions over finite fields. K. Li, C. Li, T. Helleseth, and L. Qu have recently identified in [Binary linear codes with few weights from two-to-one functions, IEEE Trans. Inf. Theory, 2021] some binary linear codes with few weights from two classes of two-to-one functions. In this paper, our ultimate objective is to expand the class of codes derived from the paper of Li <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> . by proposing larger classes of binary linear codes with few weights via generic constructions involving other known families of two-to-one functions over the finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^<i>n</i></sub> of order 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></sup> . We succeed in constructing such codes, and we also completely determine their weight distributions. The linear codes presented in this paper differ in parameters from those known in the literature. Besides, some of them are optimal concerning the well-known Griesmer bound. Notably, we prove that our codes are either optimal or almost optimal with respect to the online Database of Grassl. We next observe that the derived binary linear codes also have the minimality property for most cases. We then describe the access structures of the secret-sharing schemes based on their dual codes. Finally, we solve two problems left open in the paper by Li <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> . (more specifically, a complete solution to Problem 2 and a partial solution to Problem 1).

Linear Codes and Linear Complementary Pairs of Codes Over a Non-Chain Ring

Let [Formula: see text] be an odd prime number, [Formula: see text] for a positive integer [Formula: see text], let [Formula: see text] be the finite field with [Formula: see text] elements and [Formula: see text] be a primitive element of [Formula: see text]. We first give an orthogonal decomposition of the ring [Formula: see text], where [Formula: see text] and [Formula: see text] for a fixed integer [Formula: see text]. In addition, Galois dual of a linear code over [Formula: see text] is discussed. Meanwhile, constacyclic codes and cyclic codes over the ring [Formula: see text] are investigated as well. Remarkably, we obtain that if linear codes [Formula: see text] and [Formula: see text] are a complementary pair, then the code [Formula: see text] and the dual code [Formula: see text] of [Formula: see text] are equivalent to each other.

Construction of High-Rate Convolutional Codes Using Dual Codes

In this study, we consider techniques for searching high-rate convolutional code (CC) encoders using dual code encoders. A low-rate (R = 1/n) CC is a dual code to a high-rate (R = (n - 1)/n) CC. According to our past studies, if a CC encoder has a high performance, a dual code encoder to the CC also tends to have a good performance. However, it is not guaranteed to have the highest performance. We consider a method to obtain a high-rate CC encoder with a high performance using good dual code encoders, namely, high-performance low-rate CC encoders. We also present some CC encoders obtained by searches using our method.

Galois hulls of special Goppa codes and related codes with application to EAQECCs

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes and expurgated Goppa codes (resp., extended Goppa codes) over Fqm are GRS codes (resp., extended GRS codes) when m=1. In this paper, we investigate the Galois dual codes of a special kind of Goppa codes and related codes and provide a necessary and sufficient condition for the Galois dual codes of such codes to be Goppa codes and related codes. Then we determine the Galois hulls of the above codes. In particular, we completely characterize Galois LCD, Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such family of codes. Moreover, we apply the above results to EAQECCs.

One-generator quasi-cyclic codes and their dual codes

In this article, we study 1-generator quasi-cyclic (QC) codes over fields. QC codes of index l can be viewed as linear codes of length l. We give a description of the dual codes of 1-generator QC codes of index 2. Moreover, we obtain examples of quantum error-correcting codes (QECCs) from this study. They have better parameters than those from the online database. We consider self orthogonality conditions on such codes. We also construct some self orthogonal best known linear codes (BKLCs) linear codes, and using them we compute quantum codes. Using our symplectic study of QC codes, we are presenting three quantum codes with record-breaking parameters which improve the current record.

An open problem of k-Galois hulls and its application

The k-Galois hull of a linear code C over the finite field Fq is defined as Hullk(C)=C∩C⊥k, where q=pe, p is prime and C⊥k is the k-Galois dual code of C with 0≤k<e. In this paper, by considering an equation of the matrix rank over finite fields, one of the two problems proposed by Cao about k-Galois hull of a linear code in (arXiv:2002.12892v2, 2020) is settled. Then we construct nine new classes of MDS EAQECCs with flexible parameters. In addition, our obtained MDS EAQECCs have parameters better than the ones available in the literature.

Quantum codes from constacyclic codes over S_{k}

Let S_{k}={mathbb{F}}_{q}[u_{1},u_{2},ldots ,u_{k}]/langle u^{3}_{i}=u_{i},u_{i}u_{j}=u_{j}u_{i}=0 rangle , where 1leq i,jleq k, q=p^{m}, p is an odd prime. First, we define two new Gray maps phi _{k} and varphi _{k}, and study their Gray images. Further, we determine the structure of constacyclic codes and their dual codes, and give a necessary and sufficient conditions of constacyclic codes to contain their duals. Finally, we obtain some new quantum codes over mathbb{F}_{q} by using CSS construction, and compare the constructed codes better than the existing literature.

IBD AND RHEUMATOLOGICAL DISEASE OVERLAP IN A PREDOMINATELY AFRICAN AMERICAN CLINIC POPULATION AS COMPARED TO A NATIONAL INPATIENT DATABASE

Abstract INTRODUCTION Extraintestinal manifestations (EIM) of inflammatory bowel disease (IBD) are common and can involve nearly any organ. EIM can cause a significant impact on morbidity and quality of life and can be a challenge to physicians to manage IBD patients. Arthropathies occur in 6–53% of IBD patients and is considered the most common EIM. This study compares IBD- Rheumatological overlap in our university clinic with inpatients identified via the NIS database. We also evaluated the overlap in our clinic population concerning patient care. The objective of the study was to further characterize IBD-Rheumatological overlap by comparing our institution’s clinic population to patients identified via the NIS database. METHODS We used ICD-10 codes to identify IBD patients and further used a broad range of ICD-10 codes for rheumatological manifestation to identify overlapping patients in 2018 in our clinic population, the inpatient database population and NIS database. Incidence of coordinated care was determined in the clinic population. RESULTS In the clinic population there were 228 IBD patients and 2240 rheumatologic patients. Dual ICD-10 codes were found for 46 patients and further analysis resulted in 21 clinic patients with true overlap and recent visits (21/228 = 9.2%). For the inpatient analysis, there were 65,379 IBD patients with 3741 having a diagnosis for a rheumatologic disease (3741/65,379= 6%). In the clinic population, the majority of patients had diagnosed Crohn’s Disease (95.2%), were female (60%) and were African American (77%). The most common rheumatological manifestation was AS. While the inpatient data analysis also suggests that AS is more likely to be associated with Crohn’s Disease (0.64 % vs 0.43%), the most frequent manifestation was the general category of RA (3.54% vs 3.03%) In the National inpatient database, 22,552 patients had a diagnosis of UC out of which 81% were Caucasian, 10% African American, and 9% Hispanic. There were 38,962 patients with Crohn’s disease out of which 82% were Caucasian, 15.2% African American and 7% were Hispanic. The most common rheumatological manifestation was RA. DISCUSSION In the clinic population study where rheumatological disease was well characterized by experienced rheumatologists, the most frequent disease was AS and the primary IBD manifestation was Crohn’s disease. In the inpatient population, rheumatological disease in general was present in 6% of patients but it is unknown if more specific rheumatological evaluation was performed on the patients. The fact that the majority of clinic patients were seen by both GI and Rheumatology and were on effective therapies for their IBD and Rheumatological manifestations is encouraging. Both gastroenterologists and rheumatologists should be vigilant to identify the co-existence of these conditions and follow-up these patients for better outcomes.

Repeated-root constacyclic codes of length $ p_1p_2^t p^s $ and their dual codes

&lt;abstract&gt;&lt;p&gt;Let $ \mathbb{F}_q $ be the finite field with $ q = p^{k} $ elements, and $ p_{1}, p_{2} $ be two distinct prime numbers different from $ p $. In this paper, we first calculate all the $ q $-cyclotomic cosets modulo $ p_1p_2^t $ as a preparation for the following parts. Then we give the explicit generator polynomials of all the constacyclic codes of length $ p_1p_2^tp^s $ over $ \mathbb{F}_q $ and their dual codes. In the rest of this paper, we determine all self-dual cyclic codes of length $ p_1p_2^t p^s $ and their enumeration. This answers a question recently asked by B. Chen, H.Q.Dinh and Liu. In the last section, we calculate the case of length $ 5\ell p^{s} $ as an example.&lt;/p&gt;&lt;/abstract&gt;

On skew cyclic codes over $ M_{2}(\mathbb{F}_{2}) $

&lt;abstract&gt;&lt;p&gt;The algebraic structure of skew cyclic codes over $ M_{2} $($ \mathbb{F}_2 $), using the $ \mathbb{F}_4 $-cyclic algebra, is studied in this work. We determine that a skew cyclic code with a polynomial of minimum degree $ d(x) $ is a free code generated by $ d(x) $. According to our findings, skew cyclic codes of odd and even lengths are cyclic and $ 2 $-quasi-cyclic over $ M_{2}(\mathbb{F}_{2}) $, respectively. We provide the self-dual skew condition of Hermitian dual codes of skew cyclic codes. The generator polynomials of Euclidean dual codes are obtained. Furthermore, a spanning set of a double skew cyclic code over $ M_{2}(\mathbb{F}_{2}) $ is considered in this paper.&lt;/p&gt;&lt;/abstract&gt;

Some subfield codes from MDS codes

&lt;p style='text-indent:20px;'&gt;Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, a class of binary subfield codes is constructed from a special family of MDS codes, and their parameters are explicitly determined. The parameters of their dual codes are also studied. Some of the codes presented in this paper are optimal or almost optimal.&lt;/p&gt;

New entanglement-assisted quantum codes constructed from Hermitian LCD codes

&lt;abstract&gt;&lt;p&gt;Hermitian linear complementary dual (LCD) codes are a class of linear codes that intersect with their Hermitian dual trivially. Each Hermitian LCD code can give an entanglement-assisted quantum error-correcting code (EAQECC) with maximal entanglement. Methods of constructing Hermitian LCD codes from known codes were developed, and seven new Hermitian LCD codes with parameters $ [119,4,88]_{4} $, $ [123,4,91]_{4} $, $ [124,4,92]_{4} $, $ [136,4,101]_{4} $, $ [140,4,104]_{4} $, $ [188,4,140]_{4} $ and $ [212,4,158]_{4} $ were constructed. Seven families of Hermitian LCD codes and their related EAQECCs were derived from these codes. These new EAQECCs have better parameters than those known in the literature.&lt;/p&gt;&lt;/abstract&gt;