Dual Code Research Articles

<abstract><p>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.</p></abstract>

<abstract><p>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.</p></abstract>

<p style='text-indent:20px;'>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.</p>

<abstract><p>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.</p></abstract>

Galois dual codes are a generalization of Euclidean dual codes and Hermitian dual codes. We show that the <inline-formula><tex-math id="M910">\begin{document}$ h $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M910.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M910.png"/></alternatives></inline-formula>-Galois dual code of an algebraic geometry code <inline-formula><tex-math id="M900">\begin{document}$ C_{ {\cal{L}},F}(D,G) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M900.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M900.png"/></alternatives></inline-formula> from function field <inline-formula><tex-math id="M904">\begin{document}$ F/ \mathbb{F}_{p^e} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M904.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M904.png"/></alternatives></inline-formula> can be represented as an algebraic geometry code <inline-formula><tex-math id="M902">\begin{document}$ C_{\varOmega,F'}(\phi_{h}(D),\phi_{h}(G)) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M902.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M902.png"/></alternatives></inline-formula> from an associated function field <inline-formula><tex-math id="M903">\begin{document}$ F'/ \mathbb{F}_{p^e} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M903.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M903.png"/></alternatives></inline-formula> with an isomorphism <inline-formula><tex-math id="M600">\begin{document}$\phi_{h}:F\rightarrow F'$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M600.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M600.png"/></alternatives></inline-formula> satisfying <inline-formula><tex-math id="M700">\begin{document}$ \phi_{h}(a) = a^{p^{e-h}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M700.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M700.png"/></alternatives></inline-formula> for all <inline-formula><tex-math id="M800">\begin{document}$ a\in \mathbb{F}_{p^e} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M800.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M800.png"/></alternatives></inline-formula>. As an application of this result, we construct a family of Galois linear complementary dual maximum distance separable codes (LCD MDS codes).

A code over Gaussian or Eisenstein integer residue ring is an additive group of vectors with entries in this integer residue ring which is closed under the action of constant multiplication by the Gaussian or Eisenstein integers. In this paper, we define the dual codes for the codes over the Gaussian and Eisenstein integer residue rings, and consider the construction of the self-dual codes. Because, in the Gaussian and Eisenstein integer rings, the uniqueness of the prime element decomposition holds in the same way as the one-variable polynomial rings over finite fields and the rational integer ring, we provide an efficient construction method for self-dual code generator matrices using that of moduli. As numerical examples, for Gaussian and Eisenstein integer rings, we enumerate and construct the self-dual codes for the actual moduli when the matrix size of the generator matrices is two.

For an odd prime p ? 5, the structures of cyclic codes of length 5ps over R = Fpm + uFpm (u2 = 0) are completely determined. Cyclic codes of length 5ps over R are considered in 3 cases, namely, p ? 1 (mod 5), p ? 4 (mod 5), p ? 2 or 3 (mod 5). When p ? 1 (mod 5), a cyclic code of length 5ps over R can be expressed as a direct sum of a cyclic code and ?ps i -constacyclic codes of length ps over R, where ?ps i = ?i(pm?1)ps/10, i = 1,3,7,9. When p ? 4 (mod 5), it is equivalent to pm ? 1 (mod 5) when m is even and pm ? 4 (mod 5) when m is odd. If pm ? 1 (mod 5) when m is even, then a cyclic code of length 5ps over R can be obtained as a direct sum of a cyclic code and ?ps i -constacyclic codes of length ps over R, where ?psi = ?i(pm?1)ps/10, i = 1,3,7,9. If pm ?/ 4 (mod 5) when m is odd, then a cyclic code of length 5ps over R can be expressed as a direct sum of a cyclic code of length ps over R and an ?1 and ?2-constacyclic code of length 2ps over R, for some ?1, ?2 ? Fpm\{0}. If p ? 2 or 3 (mod 5) such that pm ?/ 1 (mod 5), then a cyclic code of length 5ps over R can be expressed as C1 ? C2, where C1 is an ideal of R[x]/?xps?1? and C2 is an ideal of R[x]/(x4+x3+x2+x+1)ps ?. We also investigate all ideals of R[x]/?(x4+x3+x2+x+1)ps ? to study detail structure of a cyclic code of length 5ps over R. In addition, dual codes of all cyclic codes of length 5ps over R are also given. Furthermore, we give the number of codewords in each of those cyclic codes of length 5ps over R. As cyclic and negacyclic codes of length 5ps over R are in a one-by-one equivalent via the ring isomorphism x ? ?x, all our results for cyclic codes hold true accordingly to negacyclic codes.

In this article, we investigate the formation of reversible cyclic codes (i.e., its codewords forms a symmetry) over the ring S=F2+uF2+u2F2, where u3=0. We find a unique set of generators for cyclic codes over S and classify reversible cyclic codes to their generators. The dual reversible cyclic codes are studied as well. Moreover, we provide some examples of reversible cyclic codes.

In this paper, we investigate the structure and properties of skew negacyclic codes and skew quasi-negacyclic codes over the ring [Formula: see text] Some structural properties of [Formula: see text] are discussed, where [Formula: see text] is an automorphism of [Formula: see text] A skew quasi-negacyclic code of length [Formula: see text] with index [Formula: see text] over [Formula: see text] is viewed both as in the conventional row circulant form and also as an [Formula: see text]-submodule of [Formula: see text], where [Formula: see text] is the Galois extension ring of degree [Formula: see text] over [Formula: see text] and [Formula: see text] is an automorphism of [Formula: see text] A sufficient condition for one generator skew quasi-negacyclic codes to be free is determined. Some distance bounds for free one generator skew quasi-negacyclic codes are discussed. Furthermore, given the decomposition of a skew quasi-negacyclic code, we provide the decomposition of its dual code. As a result, a characterization of self-dual skew quasi-negacyclic codes over [Formula: see text] is provided. By using computer search we obtained a number of new linear codes over [Formula: see text] from skew negacyclic and skew quasi-negacyclic codes over [Formula: see text].

The binary primitive BCH codes are cyclic and are constructed by choosing a subset of the cyclotomic cosets. Which subset is chosen determines the dimension, the minimum distance and the weight distribution of the BCH code. We construct possible BCH codes and determine their coderate, true minimum distance and the non-equivalent codes. A particular choice of cyclotomic cosets gives BCH codes which are, extended by one bit, equivalent to Reed-Muller codes, which is a known result from the sixties. We show that BCH codes have possibly better parameters than Reed-Muller codes, which are related in recent publications to polar codes. We study the decoding performance of these different BCH codes using information set decoding based on minimal weight codewords of the dual code. We show that information set decoding is possible even in case of a channel without reliability information since the decoding algorithm inherently calculates reliability information. Different BCH codes of the same rate are compared and different decoding performances and complexity are observed. Some examples of hard decision decoding of BCH codes have the same decoding performance as maximum likelihood decoding. All presented decoding methods can possibly be extended to include reliability information of a Gaussian channel for soft decision decoding. We show simulation results for soft decision list information set decoding and compare the performance to other methods.

The minimum weight of the code generated by the incidence matrix of points versus lines in a projective plane has been known for over 50 years. Surprisingly, finding the minimum weight of the dual code of projective planes of non-prime order is still an open problem, even in the Desarguesian case. In this paper, we focus on the case of projective planes of order p^2, where p is prime, and we link the existence of small weight code words in the dual code to the existence of embedded subplanes and antipodal planes. In the Desarguesian case, we can exclude such code words by showing a more general result that no antipodal plane of order at least 3 can be embedded in a Desarguesian projective plane. Furthermore, we use combinatorial arguments to rule out the existence of code words in the dual code of points and lines of an arbitrary projective plane of order p^2, p prime, of weight at most 2p^2-2p+4 using more than two symbols. In particular, this leads to the result that the dual code of the Desarguesian projective plane {{,textrm{PG},}}(2,p^2), pge 5, has minimum weight at least 2p^2-2p+5.

Galois LCD codes over mixed alphabets

Let R be a Galois ring, GR(pn,r), of characteristic pn and of order pnr. In this article, we study cyclic codes of arbitrary length, N, over R. We use discrete Fourier transform (DFT) to determine a unique representation of cyclic codes of length, N, in terms of that of length, ps, where s=vp(N) and vp are the p-adic valuation. As a result, Hamming distance and dual codes are obtained. In addition, we compute the exact number of distinct cyclic codes over R when n=2.

The description of the developed algorithm for optimal symbol-by-symbol reception of signal structures based on frequency-efficient signals with two-dimensional «constellations» intensively used in applications and block correction codes in non-binary Galois fields is given. It is shown that symbol-by-symbol reception minimizes the error probability for code symbol or for information bit in contrast to the well-known maximum likelihood rule which minimizes the error-probability for code word. It is proved that the basis of this decoding algorithm is the spectral transform in the Walsh-Hadamard basis and the resulting complexity of the algorithm is determined by the dimension of the dual code which makes it promising for block correction codes with a high code rate (with low redundancy). The study of the error-performance characteristics (probability of erroneous reception for information bit) of the considered symbol-by-symbol reception algorithm was carried out by simulating it for signal structures based on signals with multilevel phase shift keying (PSK-16 signals) and on signals with quadrature amplitude manipulation (QAM-16 signals) and on simple correcting code with a general parity check in non-binary Galois field of size 16. It is shown that the use of the symbol-by-symbol reception algorithm provides an energy gain for the bit error probability from 0.00001 to 1.25…1.5 in relation to the considered signals without coding. As the bit error probability decreases the energy gains increase.

Hermitian duality of left dihedral codes over finite fields

Additive complementary dual codes over $$\mathbb {F}_4$$

In 1991, Wei proved a duality theorem that established an interesting connection between the generalized Hamming weights of a linear code and those of its dual code. Wei’s duality theorem has since been extensively studied from different perspectives and extended to other settings. In this paper, we re-examine Wei’s duality theorem and its various extensions, henceforth referred to as Wei-type duality theorems, from a new Galois connection perspective. Our approach is based on the observation that the generalized Hamming weights and the dimension/length profiles of a linear code form a Galois connection. The central result of this paper is a general Wei-type duality theorem for two Galois connections between finite subsets of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}$ </tex-math></inline-formula> , from which all the known Wei-type duality theorems can be recovered. As corollaries of our central result, we prove new Wei-type duality theorems for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$w$ </tex-math></inline-formula> -demi-matroids defined over finite sets and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$w$ </tex-math></inline-formula> -demi-polymatroids defined over modules with a composition series, which further allows us to unify and generalize all the known Wei-type duality theorems established for codes endowed with various metrics.

Let A be a ring with identity, σ a ring endomorphism of A that maps the identity to itself, δ a σ -derivation of A , and consider the skew-polynomial ring A X ; σ , δ . When A is a finite field, a Galois ring, or a general ring, some fairly recent literature used A X ; σ , δ to construct new interesting codes (e.g., skew-cyclic and skew-constacyclic codes) that generalize their classical counterparts over finite fields (e.g., cyclic and constacyclic linear codes). This paper presents results concerning monic principal skew codes, called herein monic principal f , σ , δ -codes, where f ∈ A X ; σ , δ is monic. We provide recursive formulas that compute the entries of both a generator matrix and a control matrix of such a code C . When A is a finite commutative ring and σ is a ring automorphism of A , we also give recursive formulas for the entries of a parity-check matrix of C . Also, in this case, with δ = 0 , we present a characterization of monic principal σ -codes whose dual codes are also monic principal σ -codes, and we deduce a characterization of self-dual monic principal σ -codes. Some corollaries concerning monic principal σ -constacyclic codes are also given, and a good number of highlighting examples is provided.

Constructions of projective linear codes by the intersection and difference of sets

Construction of self dual codes from graphs