Dual Code Research Articles

An Algorithm for Finding Self-Orthogonal and Self-Dual Codes Over Gaussian and Eisenstein Integer Residue Rings Via Chinese Remainder Theorem

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.

Cyclic codes of length 5ps over Fpm + uFpm and their duals

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.

Reversible Cyclic Codes over F2+uF2+u2F2

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.

On some skew codes over ℤq + uℤq

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].

On Hard and Soft Decision Decoding of BCH Codes

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.

Embedded antipodal planes and the minimum weight of the dual code of points and lines in projective planes of order p^2

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

In this work we give a characterization of Galois Linear Complementary Dual codes and Galois-invariant codes over mixed alphabets of finite chain rings, which leads to the study of the Gray image of FpFp[θ]-linear codes, where p∈{2;3} and θ≠θ2=0 that provides LCD codes over Fp.

A Unique Representation of Cyclic Codes over GR(pn,r)

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.

АЛГОРИТМ ОПТИМАЛЬНОГО ПОСИМВОЛЬНОГО ПРИЕМА ЧАСТОТНО-ЭФФЕКТИВНЫХ СИГНАЛОВ С КОРРЕКТИРУЮЩИМИ КОДАМИ В НЕДВОИЧНЫХ ПОЛЯХ ГАЛУА

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

Let Fq2 be the finite field of q2 elements, where q is a power of a prime number, and let D2n=〈x,y|xn=1,y2=1,yxy=xn−1〉 be the dihedral group of 2n elements. Left ideals of the group algebra Fq2[D2n] are known as left dihedral codes over Fq2 of length 2n, and abbreviated as left D2n-codes. Let gcd(n,q)=1. In this paper, we give an explicit representation for the Hermitian dual code and the Hermitian hull of every left D2n-code over Fq2. On this basis, we determine all distinct Hermitian self-dual left D2n-codes, Hermitian linear complementary dual (LCD) left D2n-codes, and Hermitian self-orthogonal left D2n-codes over Fq2, respectively. Then we provide an explicit representation and a precise enumeration for these three subclasses of left D2n-codes. As an application, we provide several illustrative examples for obtaining Hermitian self-dual and Hermitian LCD left D2n-codes respectively.

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

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

A Galois Connection Approach to Wei-Type Duality Theorems

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.

Characterizations and Properties of Monic Principal Skew Codes over Rings

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

Projective linear codes are a special class of linear codes whose dual codes have minimum distance at least 3. Projective linear codes with only a few weights are useful in authentication codes, secret sharing schemes, data storage systems and so on. In this paper, two constructions of q-ary linear codes are presented with defining sets given by the intersection and difference of two sets. These constructions produce several families of new projective two-weight or three-weight linear codes. As applications, our projective codes can be used to construct secret sharing schemes with interesting access structures, strongly regular graphs and association schemes with three classes.

Construction of self dual codes from graphs

Construction of self dual codes from graphs

Additive cyclic complementary dual codes over [formula omitted

An additive cyclic code of length n over F4 can be defined equivalently as an F2[x]/〈xn+1〉-submodule of F4[x]/〈xn+1〉. In this paper we study additive cyclic and complementary dual codes of odd length over F4 with respect to the trace Hermitian inner product and the trace Euclidean inner product. We characterize subfield subcodes and trace codes of these codes by their generators as binary cyclic codes.

Additive Complementary Dual Codes From Group Characters

Additive codes have become an increasingly important topic in algebraic coding theory due to their applications in quantum error-correction and quantum computing. Linear Complementary Dual (LCD) codes play an important role for improving the security of information against certain attacks. Motivated by these facts, we define additive complementary dual codes (ACD for short) over a finite abelian group in terms of an arbitrary duality on the ambient space and examine their properties. We show that the best minimum weight of ACD codes is always greater than or equal to the best minimum weight of LCD codes of the same size and that this inequality is often strict. We give some matrix constructions for quaternary ACD codes from a given quaternary ACD code and also from a given binary self-orthogonal code. Moreover, we construct an algorithm to determine if a given quaternary additive code is an ACD code with respect to all possible symmetric dualities. We also determine the largest minimum distance of quaternary ACD codes for lengths <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n \leq 10$ </tex-math></inline-formula> . The obtained codes are either optimal or near optimal according to Bierbrauer <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> +. (2009).

GF(q) LDPC encoder and decoder FPGA implementation using group shuffled belief propagation algorithm

This paper presents field programmable gate array (FPGA) exercises of the GF(q) low-density parity-check (LDPC) encoder and interpreter utilizing the group shuffled belief propagation (GSBP) algorithm are presented in this study. For small blocks, non-dual LDPC codes have been shown to have a greater error correction rate than dual codes. The reduction behavior of non-binary LDPC codes over GF (16) (also known as GF(q)-LDPC codes) over the additive white Gaussian noise (AWGN) channel has been demonstrated to be close to the Shannon limit and employs a short block length (N=600 bits). At the same time, it also provides a non-binary LDPC (NB-LDPC) code set program. Furthermore, the simplified bubble check treasure event count is implemented through the use of first in first out (FIFO), which is based on an elegant design. The structure of the interpreter and the creation of the residential area he built were planned in very high speed integrated circuit (VHSIC) hardware description language (VHDL) and simulated in MODELSIM 6.5. The combined output of the Cyclone II FPGA is combined with the simulation output.

Some hypersurfaces over finite fields, minimal codes and secret sharing schemes

Linear error-correcting codes can be used for constructing secret sharing schemes; however, finding in general the access structures of these secret sharing schemes and, in particular, determining efficient access structures is difficult. Here we investigate the properties of certain algebraic hypersurfaces over finite fields, whose intersection numbers with any hyperplane only takes a few values; these varieties give rise to q-divisible linear codes with at most 5 weights. Furthermore, for q odd, these codes turn out to be minimal and we characterize the access structures of the secret sharing schemes based on their dual codes. Indeed, the secret sharing schemes thus obtained are democratic, that is each participant belongs to the same number of minimal access sets and can easily be described.

On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and Their Consequences

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field $\mathrm{GF}(q)$ with length $q+1$, which are closely related to the action of the projective general linear group of degree two on the projective line. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$-ary BCH codes with length $q+1$, and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of $\mathrm{PGL} (2, p^m)$-invariant codes over $\mathrm{GF} (p^h)$ is completed. As an application of this result, the $p$-ranks of all incidence structures invariant under the projective general linear group $\mathrm{ PGL }(2, p^m)$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a $3$-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms.