Lee Distance Research Articles

Density of k-ary words with 0, 1, 2 - error overlaps

An overlap, or border, of a word is a prefix that is equal to the suffix of the same length. An overlap with q errors is a prefix which has distance q from the suffix of the same length; here, 0-error overlaps are classic ones. Unbordered, or bifix-free, words are a central notion in combinatorics on words and have a prominent role in many related areas, such as pattern matching or frame synchronization. On the other hand, words with 2-error overlaps arose as a characterization of isometric words, a notion recently introduced in the framework of hypercubes and their isometric subgraphs. This paper investigates the density of words with 0, 1, 2-error overlaps, where the words are taken over a generic k-ary alphabet, k≥2, and the distance they refer to is the Hamming or the Lee distance. Estimates on the limit density values are provided and compared in the case of binary and quaternary alphabets.

Characterization of some specific lee distance codes over finite rings

Codes over integer rings equipped with the Lee metric are gaining attention among researchers recently. In this paper, we investigate some specific linear codes over integer rings and characterize their dimensions and Lee weights using greatest common divisors (GCD). Furthermore, we construct equidistant Lee codes over [Formula: see text] which have applications in error detection and error correction.

On reversibility problem in DNA bases over a class of rings

Let R k = Z 4 [ u 1 , u 2 , … , u k ] / ⟨ u i 2 − u i , u i u j − u j u i ⟩ be a non-chain ring of characteristic 4, where 1 ≤ i , j ≤ k and k ≥ 1 . In this article, we discuss reversible cyclic codes of odd lengths over the ring R k . We construct bijections between the elements of the ring R k and DNA- 2 k bases for k = 1, 2 in such a manner that the reversibility problem is solved. Employing these bijections, reversible complement cyclic codes of odd lengths are generated. Furthermore, we construct a Gray map Φ : R k n → Z 4 2 k n and as an application of the Gray map Φ, we obtain the GC-content of cyclic codes of arbitrary odd length over the ring R k . Meanwhile, we provide some examples of reversible cyclic codes of odd lengths over the ring R k for different values of k, and also obtain the Lee distances of these codes.

Computing the index of non-isometric k-ary words with Hamming and Lee distance

Isometric words are those words whose occurrence as a factor in a transformation of a word u in a word v can be avoided, while preserving the minimal length of the transformation. Such minimal length refers to a distance between u and v. In the literature, isometric words have been considered with respect to the Hamming distance and the Lee distance; the former especially for binary words, while the latter for k-ary words, with k ⩾ 2. Ham- and Lee- isometric words have been characterized in terms of their overlaps with errors. In this paper, we give algorithms to decide whether a word f, of length n, is Ham- or Lee-isometric and provide evidence of the possible non-isometricity by returning a pair of words of minimal length whose transformation cannot avoid the factor f. Such a pair of words is called a pair of witnesses and the minimal length of the witnesses is called the index of f. The algorithms run in O ( n ) time with a preprocessing of O ( n ) time and space to construct a data structure that allows answering LCA queries on the suffix tree of f in constant time. The correctness of the algorithms lies on some theoretical results on the index and the witnesses of a word that are here presented. The investigation on the index is completed by the characterization of words with minimum/maximum index. All the results are shown referring to both Hamming and Lee distance.

Duadic codes over Fp + uFp + vFp + uvFp

Duadic codes constitute a well-known class of cyclic codes. In this paper, we study the structure of duadic codes of length n over the ring R = Fp + uFp + vFp + uvFp, u2 = v2 = 0, uv = vu, where p is prime and (n, p) = 1. These codes have been studied here in the setting of abelian codes over R, and we have used Fourier transform and idempotents to study them. We have characterized abelian codes over R by studying their torsion and residue codes. It is shown that the Gray image of an abelian code of length n over R is a binary abelian code of length 4n. Conditions for self-duality and self-orthogonality of duadic codes over R are derived. Some conditions on the existence of self-dual augmented and extended codes over R are presented. We have also studied Type II self-dual augmented and extended codes over R. Some results related to theminimum Lee distances of duadic codes over R are presented. We have also presented a sufficient condition for abelian codes of the same length over R to have the same minimum Hamming distance. Some optimal binary linear codes of length 36 and ternary linear codes of length 16 have been obtained as Gray images of duadic codes of length 9 and 4, respectively, over R using the computational algebra system Magma.

Repeated root cyclic codes over $\mathbb {Z}_{p^{2}}+u\mathbb {Z}_{p^{2}}$ and their Lee distances

Repeated root cyclic codes over $\mathbb {Z}_{p^{2}}+u\mathbb {Z}_{p^{2}}$ and their Lee distances

Lee distance of cyclic and (1 + uγ)-constacyclic codes of length 2s over [formula omitted

Let m be a positive integer, and γ be a non-zero element of F2m. The Lee distance of all cyclic codes and (1+uγ)-constacyclic codes of length 2s over F2m+uF2m is completely determined. In particular, it is shown that the Lee distance of such codes is independent of the choice of a trace orthogonal basis of F2m.

Lee Distance of (4z – 1)-Constacyclic Codes of Length 2 s Over the Galois Ring GR(2 a ,m)

Let a and m be positive integers. The definition of Lee weights over ℤ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2a </sub> is generalized to its m-degree Galois extension GR(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> , m) using a nonzero element ξ in GR(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> , m) of order 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> - 1. For z ∈ GR(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> , m), the minimum Lee weights of (4z-1)-constacyclic codes over GR(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> , m) of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> are determined and shown to be independent of the choice of ξ. Using this, we construct some optimal binary codes of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> as Gray images of certain (4z - 1)-constacyclic codes over GR(4, m) and obtain some maximum distance separable (MDS) codes with respect to the Lee distance of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> over GR(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> , m).

Decoding of cyclic codes over the ring $$F_2+uF_2+u^2F_2$$

Udaya and Bonnecaze (IEEE Trans Inf Theory 45:2148–2157, 1999) presented a decoding algorithm for cyclic codes of odd length over the ring $$F_2+u F_2$$ . In this study, a simpler approach for decoding cyclic codes with odd length over this ring is proposed. The structure of cyclic codes of odd length over the ring $$R=F_2+uF_2+u^2F_2$$ , where $$u^3=0,$$ is given. A Gray map which is both an isometry and a weight-preserving map from $$R^n$$ to $${F_2}^{4n}$$ is defined and with the use of proposed Gray map, a BCH-like bound for the Lee distance of codes over R is given. Finally, a decoding algorithm is suggested for cyclic codes over R.

Correction to: Asymptotic properties of Lee distance

The statement of Theorem 3 (p. 397) in (Nikolov and Stoimenova 2019) should be corrected to.

On the Generalized Concatenated Construction for Codes in $${L_1}$$ and Lee Metrics

We consider a generalized concatenated construction for error-correcting codes over the q-ary alphabet in the modulus metric L1 and Lee metric L. Resulting codes have arbitrary length, arbitrary distance (independently of the alphabet size), and can correct both independent errors and error bursts in both metrics. In particular, for any length 2m we construct codes over $$\mathbb{Z}_4$$ with Lee distance 4 which under the Gray mapping yield extended binary perfect codes of length 2m+1 (with code distance 4). We construct codes over $$\mathbb{Z}_4$$ of length n with Lee distance n which under the Gray mapping yield Hadamard matrices of order 2n (under the additional condition that an Hadamard matrix of order n exists). The constructed new codes in the Lee metric are often better in their parameters than previously known ones; in particular, they are essentially better than previously constructed Astola codes.

Codes from incidence matrices of some regular graphs

We examine the [Formula: see text]-ary linear codes with respect to Lee metric from incidence matrix of the Lee graph with vertex set [Formula: see text] and two vertices being adjacent if their Lee distance is one. All the main parameters of the codes are obtained as [Formula: see text] if [Formula: see text] is odd and [Formula: see text] if [Formula: see text] is even. We examine also the [Formula: see text]-ary linear codes with respect to Hamming metric from incidence matrices of Desargues graph, Pappus graph, Folkman graph and the main parameters of the codes are [Formula: see text], respectively. Any transitive subgroup of automorphism groups of these graphs can be used for full permutation decoding using the corresponding codes. All the above codes can be used for full error correction by permutation decoding.

Self-dual codes over $\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})$

In this study we consider Euclidean and Hermitian self-dual codes over the direct product ring $\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})$ where v2 = v. We obtain some theoretical outcomes about self-dual codes via the generator matrices of free linear codes over $\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})$ . Also, we obtain upper bounds on the minimum distance of linear codes for both the Lee distance and the Gray distance. Moreover, we find some free Euclidean and free Hermitian self-dual codes over $\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})$ via some useful construction methods.

On the Covering Radius of Codes over Z p k

In this correspondence, we investigate the covering radius of various types of repetition codes over Z p k ( k ≥ 2 ) with respect to the Lee distance. We determine the exact covering radius of the various repetition codes, which have been constructed using the zero divisors and units in Z p k . We also derive the lower and upper bounds on the covering radius of block repetition codes over Z p k .

The large capacity embedding algorithm for H.264/AVC intra-prediction mode video steganography based on linear block code over Z4

The H.264/AVC intra prediction video steganography based on (n,k) linear block code over ring Z4 is proposed by this paper. On the basis of (n,k) linear block code over Z4, the property of addition of abelian groups and multiplicative commutative semigroup are utilized to generate the steganography code. Lee distance of linear codes over ring Z4 is researched, and it makes the equation has unique solution. Furthermore, the optimal performance of the proposed code is proved. The code performance with high embedding rate whose upper bound of embedding rate is reached to 1.33bpp, and the code is applied to video steganography based on intra-prediction mode (IPM-based video steganography). It obtains better performances of security and anti-attack. The average distortion is decreased to 50% and the detection accuracy is decreased to 3-5% with identical embedding rate. Experimental results show that the performance of the proposed scheme is superior to those of the former IPM-based schemes.

Do non-free LCD codes over finite commutative Frobenius rings exist?

In this paper, we clarify some aspects of LCD codes in the literature. We first prove that non-free LCD codes do not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD codes over a finite commutative Frobenius ring. We later show that a free constacyclic code over a finite chain ring is an LCD code if and only if it is reversible, and also provide a necessary and sufficient condition for a constacyclic code to be reversible. We illustrate the minimum Lee distance of LCD codes over some finite commutative chain rings with examples. We found some new optimal cyclic codes over $${\mathbb {Z}}_4$$ of different lengths which are LCD codes using computer algebra system MAGMA.

Semidefinite programming bounds for Lee codes

For q,n,d∈N, let AqL(n,d) denote the maximum cardinality of a code C⊆Zqn with minimum Lee distance at least d, where Zq denotes the cyclic group of order q. We consider a semidefinite programming bound based on triples of codewords, which bound can be computed efficiently using symmetry reductions, resulting in several new upper bounds on AqL(n,d).The technique also yields an upper bound on the independent set number of the nth strong product power of the circular graph Cd,q, which number is related to the Shannon capacity of Cd,q. Here Cd,q is the graph with vertex set Zq, in which two vertices are adjacent if and only if their distance (mod q) is strictly less than d. The new bound does not seem to improve significantly over the bound obtained from Lovász theta-function, except for very small n.

On codes over ℤp2 and its covering radius

This paper gives lower and upper bounds on the covering radius of codes over [Formula: see text] with respect to Lee distance. We also determine the covering radius of various repetition codes over [Formula: see text]

Designing Uniform Computer Sequential Experiments with Mixture Levels Using Lee Discrepancy

Computer experiments are constructed to simulate the behavior of complex physical systems. Uniform designs have good performance in computer experiments from several aspects. In practical use, the experimenter needs to choose a small size uniform design at the beginning of an experiment due to a limit of time, budget, resources, and so on, and later conduct a follow up experiment to obtain precious information about the system, that is, a sequential experiment. The Lee distance has been widely used in coding theory and its corresponding discrepancy is an important measure for constructing uniform designs. This paper proves that all the follow up designs of a uniform design are uniform and at least two of them can be used as optimal follow up experimental designs. Thus, it is not necessary that the union of any two uniform designs yields a uniform sequential design. Therefore, this article presents a theoretical justification for choosing the best follow up design of a uniform design to construct a uniform sequential design that involves a mixture of ω ≥ 1 factors with βk ≥ 2, 1 ≤ k ≤ ω levels. For illustration of the usage of the proposed results, a closer look is given at using these results for the most extensively used six particular cases, three symmetric and three asymmetric designs, which are often met in practice.

On covering radius of codes over ℤ2p

In this paper, we gives lower and upper bounds on the covering radius of codes over [Formula: see text], where [Formula: see text] is a prime integer with respect to Lee distance. We also determine the covering radius of various Repetition codes over [Formula: see text], where [Formula: see text] is a prime integer.