Narrow Results

In this paper, we consider the challenge of ensuring secure communication in block-fading wiretap channels using encoding techniques. We investigate the coding for block-fading wiretap channels using stacked lattice codes constructed over completely real number fields, which is a well-established technique. In this paper, we consider degree-four complex multiplication field ${𝒦}=\mathbb{Q}({\varsigma }_8)$ over $\mathbb{Q}$. We employ binary codes to generate a lattice over $\mathbb{Q}({\varsigma }_8)$, which is subsequently used to form an integral lattice. The resulting integral lattice can be effectively applied to enhance the security of communication within block-fading wiretap channels.

In this paper, we present an isomorphism that connects a code of length $n$ over ${𝔽}_{q^m}$ to a code of length $mn$ over ${𝔽}_q$. We define criteria for identifying linear complementary pair (LCP) and linear complementary dual (LCD) codes by utilizing this isomorphism. We also introduce the notion of $\ell$-LCP of codes into the LCP framework and give a necessary and sufficient condition for their characterization. We also provide a necessary and sufficient condition for the existence of an maximum distance separable (MDS) $\ell$-LCP of codes. These insights into the structural properties of codes offer promising avenues for their application in various fields, particularly in the realm of finite fields.

Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial $t$-designs have attracted lots of research interest for decades. The interplay between coding theory and $t$-designs started many years ago. It is generally known that $t$-designs can be used to derive linear codes over any finite field, and that the supports of all codewords with a fixed weight in a code also may hold a $t$-design. In this paper, we first construct a class of linear codes from cyclic codes related to Dembowski-Ostrom functions. By using exponential sums, we then determine the weight distribution of the linear codes. Finally, we obtain infinite families of $2$-designs from the supports of all codewords with a fixed weight in these codes. Furthermore, the parameters of $2$-designs are calculated explicitly.

Let K be a finite field and let X be a subset of a projective space, over the field K, which is parametrized by monomials arising from the edges of a clutter. We show some estimates for the degree-complexity, with respect to the revlex order, of the vanishing ideal I(X) of X. If the clutter is uniform, we classify the complete intersection property of I(X) using linear algebra. We show an upper bound for the minimum distance of certain parametrized linear codes along with certain estimates for the algebraic invariants of I(X).

We propose a decoding algorithm for the (u | u + v)-construction that decodes up to half of the minimum distance of the linear code. We extend this algorithm for a class of matrix-product codes in two different ways. In some cases, one can decode beyond the error-correction capability of the code.

The extension problem for linear codes over modules with respect to Hamming weight was already settled in [J. A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc.136 (2008) 699–706; Foundations of linear codes defined over finite modules: The extension theorem and MacWilliams identities, in Codes Over Rings, Series on Coding Theory and Cryptology, Vol. 6 (World Scientific, Singapore, 2009), pp. 124–190]. A similar problem arises naturally with respect to symmetrized weight compositions (SWC). In 2009, Wood proved that Frobenius bimodules have the extension property (EP) for SWC. More generally, in [N. ElGarem, N. Megahed and J. A. Wood, The extension theorem with respect to symmetrized weight compositions, in 4th Int. Castle Meeting on Coding Theory and Applications (2014)], it is shown that having a cyclic socle is sufficient for satisfying the property, while the necessity remained an open question. Here, landing in midway, a partial converse is proved. For a (not small) class of finite module alphabets, the cyclic socle is shown necessary to satisfy the EP. The idea is bridging to the case of Hamming weight through a new weight function.

Note: All rings are finite with unity, and all modules are finite too. This may be re-emphasized in some statements. The convention for left homomorphisms is that inputs are to the left.

Let ${𝔽}_{p^m}$ be a finite field of cardinality $p^m$, where $p$ is an odd prime, $k,\lambda$ be positive integers satisfying $\lambda \ge 2$, and denote ${𝒦}={𝔽}_{p^m}[x]/〈f{(x)}^{\lambda p^k}〉$, where $f(x)$ is an irreducible polynomial in ${𝔽}_{p^m}[x]$. In this note, for any fixed invertible element $\omega \in {{𝒦}}^{\times }$, we present all distinct linear codes $S$ over ${𝒦}$ of length $2$ satisfying the condition: $(\omega f{(x)}^{p^k}{a}_1,a_0)\in S$ for all $(a_0,a_1)\in S$. This conclusion can be used to determine the structure of $(\delta +\alpha u^2)$-constacyclic codes over the finite chain ring ${𝔽}_{p^m}[u]/〈u^{2\lambda }〉$ of length $n{p}^k$ for any positive integer $n$ satisfying $\mathrm{\text{gcd}}(p,n)=1$.

This paper considers a new alphabet set, which is a ring that we call ${𝔽}_{4}R$, to construct linear error-control codes. Skew cyclic codes over this ring are then investigated in details. We define a nondegenerate inner product and provide a criteria to test for self-orthogonality. Results on the algebraic structures lead us to characterize ${𝔽}_{4}R$-skew cyclic codes. Interesting connections between the image of such codes under the Gray map to linear cyclic and skew-cyclic codes over ${𝔽}_4$ are shown. These allow us to learn about the relative dimension and distance profile of the resulting codes. Our setup provides a natural connection to DNA codes where additional biomolecular constraints must be incorporated into the design. We present a characterization of $R$-skew cyclic codes which are reversible complement.

In this paper, an intrinsic description of some families of linear codes with symmetries is given, showing that they can be described more generally as quasi-group codes, that is, as linear codes allowing a group of permutation automorphisms which acts freely on the set of coordinates. An algebraic description, including the concatenated structure, of such codes is presented. This allows to construct quasi-group codes from codes over rings, and vice versa. The last part of the paper is dedicated to the investigation of self-duality of quasi-group codes.

Theory of wavelet transform is a powerful tool for image and video processing. Mathematical concepts of wavelet transform and filter bank have been studied carefully in many works. This work presents application of new construction of linear and robust codes based on wavelet decomposition and its application in ADV612 chips. We present the model of the error-coding scheme that allows to detect errors in the ADV612 chips with high probability. In our work, we will show that developed and presented scheme of protection drastically improves the resistance of ADV612 chips to malfunctions and errors.

As an alternative to the usual key generation by two-way communication in schemes for quantum cryptography, we consider codes for key generation by one-way communication. We study codes that could be applied to the raw key sequences that are ideally obtained in recently proposed scenarios for quantum key distribution, which can be regarded as communication through symmetric four-letter channels.

We give a construction for quantum codes from linear and cyclic codes over . We derive Hermitian self-orthogonal codes over as Gray images of linear and cyclic codes over . In particular, we use two binary codes associated with a cyclic code over of odd length to determine the parameters of the corresponding quantum code.

The asymmetric CSS construction is extended to the Hermitian dual case. New infinite families of quantum symmetric and asymmetric codes are constructed. In particular, new quantum codes are obtained from binary BCH codes and MDS codes. These codes have known minimum distances and thus the relationship between the rate gain and minimum distance is given explicitly. The codes obtained are shown to have parameters better than those of previous codes. A number of known codes are special cases of the codes given here.

Let R = F₃ + vF₃ be a finite commutative ring, where v² = 1. It is a finite semi-local ring, not a chain ring. In this paper, we give a construction for quantum codes from cyclic codes over R. We derive self-orthogonal codes over F₃ as Gray images of linear and cyclic codes over R. In particular, we use two codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code.

Lee weight is more appropriate for some practical situations than Hamming weight as it takes into account the magnitude of each digit of the word. In this paper, we obtain a sufficient bound over the number of parity check digits for codes detecting burst errors and also for codes correcting burst errors with Lee weight.

There are three standard weight functions on a linear code viz. the Hamming weight, Lee weight and Euclidean weight. The Euclidean weight function is useful in connection with the lattice constructions, where the minimum norm of vectors in the lattice is related to the minimum Euclidean weight of the code. In this paper, we obtain Singleton's bound for Euclidean codes and introduce maximum Euclidean square distance separable (MESDS) codes.

In this paper, we first obtain a sufficient condition for a linear code to have minimum square Euclidean distance at least d². Using this condition, we then obtain a sufficient lower bound for random error correction and a bound on the maximum dimension k of Euclidean codes having minimum square distance at least d².

We obtain a construction upper bound over the number of parity check digits for Lee weight codes correcting random errors and simultaneously detecting CT-burst errors.

The Lee weight is more appropriate for some practical situations than the Hamming weight as it takes into account of the magnitude of each digit of the word. In this paper, we obtain a sufficient condition over the number of parity check digits for codes correcting simultaneously random and burst errors with Lee weight consideration. This sufficient condition is an extension of the Varshamov-Gilbert-Sacks bound for codes correcting simultaneously random and burst errors with Lee weight constraint.

Recently, Venkatesh improved the best known lower bound for lattice sphere packings by a factor $loglogn$ for infinitely many dimensions $n$. Here, we prove an effective version of this result, in the sense that we exhibit, for the same set of dimensions, finite families of lattices containing a lattice reaching this bound. Our construction uses codes over cyclotomic fields, lifted to lattices via Construction A. We also prove a similar result for families of symplectic lattices.