Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: A Correction to this paper has been published: https://doi.org/10.1007/s12095-022-00602-6 PubDate: 2023-02-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Levenshtein first put forward the sequence reconstruction problem in 2001. This problem sets a model in which a sequence from some set is transmitted over multiple channels, and the decoder receives the different outputs. In this model, the sequence reconstruction problem is to find the minimum number of channels required to exactly recover the transmitted sequence. In the combinatorial context, the problem is equivalent to determining the maximum intersection between two balls of radius r, where the distance between their centers is at least d. The sequence reconstruction problem was studied for strings, permutations and so on. In this paper, we extend the study by Konstantinova et al. for reconstruction of permutations distorted by single Kendall τ-errors. While they solved the case where the transmitted permutation can be arbitrary and the erroneous patterns are distorted by at most two Kendall τ-errors, we study the setup where the transmitted permutation belongs to a permutation code of length n and the erroneous patterns are distorted by at most three Kendall τ-errors. In this scenario, it is shown that n2 − n + 1 erroneous patterns are required in order to reconstruct an unknown permutation from some permutation code of minimum Kendall τ-distance 2 or an arbitrary unknown permutation for any n ≥ 3. PubDate: 2023-02-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Multi-twisted (MT) additive codes over finite fields constitute a special class of additive codes and are generalizations of cyclic additive codes. In this paper, we study a class of 1-generator MT additive codes over finite fields. We further study the asymptotic behaviour of the rates and relative Hamming distances of a certain special subclass of 1-generator MT additive codes by using probabilistic methods. We show that there exists an asymptotically good infinite sequence of MT additive codes of length pγℓ with block length \(p^{\gamma } \to \infty\) over \(\mathbb {F}_{q^{t}}\) of rate \(0<\frac {f}{\ell t p^{d}}<1\) and relative Hamming distance at least δ, where ℓ ≥ 1 and t ≥ 2 are integers, q is a prime power, \(\mathbb {F}_{q^{t}}\) is the finite field of order qt, p is an odd prime satisfying \(\gcd (p,q)=\gcd (p,q-1)=1,\) f is the multiplicative order of q modulo p, d is the largest positive integer such that pd divides qf − 1, γ is a positive integer satisfying γ ≥ d + 1, and δ is a positive real number satisfying \(h_{q^{t}}(\delta ) < 1-\frac {1}{\ell t},\) (here \(h_{q^{t}}\) denotes the qt-ary entropy function). This shows that the family of MT additive codes over finite fields is asymptotically good. PubDate: 2023-02-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Almost complementary pairs (ACPs) are sequence pairs having zero aperiodic autocorrelation sums at all non-zero time-shifts except only one position. These are generalizations of the well-known Golay complementary pairs (GCPs). To date, known ACPs have been constructed via standard Golay sequences (GSs). In this paper, we construct novel quaternary and 16-QAM ACPs of non-power-of-two lengths based on non-standard GSs. It turns out that the peak-to-mean envelope power ratios (PMEPRs) of the proposed ACPs are upper bounded by 3.34. Due to their good correlation and low PMEPR properties, ACPs could be used as an alternative to GCPs in practical applications. PubDate: 2023-02-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Constant dimension codes (CDCs) have drawn extensive attention due to their applications in random network coding. A fundamental problem for CDCs is to explore the maximum possible cardinality Aq(n,d,k) of a set of k-dimensional subspaces in \(\mathbb {F}^{n}_{q}\) such that the subspace distance statisfies dis(U,V ) = 2k − 2 dim(U ∩ V ) ≥ d for all pairs of distinct subspaces U and V in this set. In this paper, by means of an appropriate combination of the matrix blocks from rank metric codes and small CDCs, we present three constructions of CDCs based on the generalized block inserting construction by Niu et al. in 2021. According to our constructions, we obtain 28 new lower bounds for CDCs which are better than the previously known lower bounds. PubDate: 2023-02-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract In this paper, we study Hermitian dual-containing constacyclic codes over the finite ring \(R_{r}=\mathbb {F}_{q^{2}}+{v_{1}}\mathbb {F}_{q^{2}}+\cdots +{v_{r}}\mathbb {F}_{q^{2}}\) , where q is a prime power and \({v_{i}}^{2}={v_{i}},v_{i}v_{j}=v_{j}v_{i}=0\) for 1 ≤ i,j ≤ r,i≠j. A necessary and sufficient condition is provided to determine whether a constacyclic code C over Rr is Hermitian dual-containing. Moreover, we propose a generalized Gray map ΦM to preserve the property of Hermitian dual-containing. Compared with some existing Gray maps, ΦM increases the possibility of making the minimum distance of ΦM(C) larger. As an application, some new quantum codes over \(\mathbb {F}_{q}\) are constructed from constacyclic codes over Rr. PubDate: 2023-02-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract This paper presents a direct construction of aperiodic q-ary (q is a positive even integer) even length Z-complementary pairs (ZCPs) with large zero-correlation zone (ZCZ) width using generalised Boolean functions (GBFs). The applicability of ZCPs increases with the increasing value of ZCZ width, which plays a significant role in reducing interference in a communication system with asynchronous surroundings. For q = 2, the proposed ZCPs reduce to even length binary ZCPs (EB-ZCPs). However, to the best of the authors’ knowledge, the highest ZCZ ratio for even length ZCPs which are directly constructed to date using GBFs is 3/4. In the proposed construction, we provide even length ZCPs with ZCZ ratios 5/6 and 6/7, which are the largest ZCZ ratios achieved to date through direct construction. PubDate: 2023-02-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract In this paper, two constructions of frequency-hopping sequence (FHS for short) sets with respect to aperiodic Hamming correlation (AHC for short) are presented. Based on the presented constructions, we obtain several classes of optimal or near optimal FHS sets from cyclic codes and some known FHS sets with periodic partial Hamming correlation (PPHC for short), and explicitly determine their parameters. Moreover, the parameters of the obtained FHS sets are new and have large family size. PubDate: 2023-02-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Decomposing permutation polynomials into power permutations of small algebraic facilitates the reduction of hardware area requirements when symmetric cryptographic algorithms are implemented with countermeasures to side channel attacks in mind. Our contribution in this work is twofold. Firstly, we determine such decompositions into power permutations of the field \({\mathbb F}_{2^{n}}\) of degree at most 2 for infinitely many odd n satisfying a specific property. Secondly, after improving the complexity of an existing algorithm, we use it to complete the list of known shortest decompositions of the inversion in \({\mathbb F}_{2^{n}}\) up to n = 32, and show that always considering degrees at most 3 can lead to shorter lengths or smaller area requirements. PubDate: 2023-02-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract This paper presents a formula for the dimension of Galois hulls of constacyclic codes. For this, we have arranged the irreducible factors of xn − λ over the finite field \(\mathbb {F}_{q}\) in a suitable way. Also, considering some restrictions on q, the number of constacyclic codes of length n over \(\mathbb {F}_{q}\) is calculated for a given Galois hull dimension. PubDate: 2023-02-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract We define the class of triplicate functions as a generalization of 3-to-1 functions over \(\mathbb {F}_{2^{n}}\) for even values of n. We investigate the properties and behavior of triplicate functions, and of 3-to-1 among triplicate functions, with particular attention to the conditions under which such functions can be APN. We compute the exact number of distinct differential sets of power APN functions and quadratic 3-to-1 functions; we show that, in this sense, quadratic 3-to-1 functions are a generalization of quadratic power APN functions for even dimensions, in the same way that quadratic APN permutations are generalizations of quadratic power APN functions for odd dimensions. We show that quadratic 3-to-1 APN functions cannot be CCZ-equivalent to permutations in the case of doubly-even dimensions. We compute a lower bound on the Hamming distance between any two quadratic 3-to-1 APN functions, and give an upper bound on the number of such functions over \(\mathbb {F}_{2^{n}}\) for any even n. We survey all known infinite families of APN functions with respect to the presence of 3-to-1 functions among them, and conclude that for even n almost all of the known infinite families contain functions that are quadratic 3-to-1 or are EA-equivalent to quadratic 3-to-1 functions. We also give a simpler univariate representation in the case of singly-even dimensions of the family recently introduced by Göloglu than the ones currently available in the literature. We conduct a computational search for quadratic 3-to-1 functions in even dimensions n ≤ 12. We find six new APN instances for n = 10, and the first sporadic APN instance for n = 12 since 2006. We provide a list of all known 3-to-1 APN functions for n ≤ 12. PubDate: 2023-02-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract In this paper, we first discuss linear codes over R and present the decomposition structure of linear codes over the mixed alphabet \(\mathbb {F}_{q}R\) , where \(R=\mathbb {F}_{q}+u\mathbb {F}_{q}+v\mathbb {F}_{q}+uv\mathbb {F}_{q}\) , with u2 = 1,v2 = 1, uv = vu and q = pm for odd prime p, positive integer m. Let ðœƒ be an automorphism on \(\mathbb {F}_{q}\) . Extending ðœƒ to Θ over R, we study skew (ðœƒ,Θ)-(λ,Γ)-constacyclic codes over \(\mathbb {F}_{q}R\) , where λ and Γ are units in \(\mathbb {F}_{q}\) and R, respectively. We also show that, the dual of a skew (ðœƒ,Θ)-(λ,Γ)-constacyclic code over \(\mathbb {F}_{q}R\) is a skew (ðœƒ,Θ)-(λ− 1,Γ− 1)-constacyclic code over \(\mathbb {F}_{q}R\) . We classify some self-dual skew (ðœƒ,Θ)-(λ,Γ)-constacyclic codes using the possible values of units of R. Also using suitable values of λ,ðœƒ,Γ and Θ, we present the structure of other linear codes over \(\mathbb {F}_{q} R\) . We construct a Gray map over \(\mathbb {F}_{q}R\) and study the Gray images of skew (ðœƒ,Θ)-(λ,Γ)-constacyclic codes over \(\mathbb {F}_{q}R\) . As applications of our study, we construct many good codes, among them, there are 17 optimal codes and 2 near-optimal codes. Finally, we discuss the advantages in a construction of quantum error-correcting codes (QECCs) from skew ðœƒ-cyclic codes than from cyclic codes over \(\mathbb {F}_{q}\) . PubDate: 2023-02-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract In this paper, we study the symbol-pair weight distribution of some irreducible cyclic codes. We define generalized cyclotomic numbers and provide some of their properties. Using a character sum, we establish a relationship among the symbol-pair weight of irreducible cyclic codes, generalized cyclotomic numbers and the Gaussian periods. Moreover, we determine the symbol-pair weight distribution of some classes of irreducible cyclic codes. PubDate: 2023-02-01

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract MDS constacyclic codes over finite fields are important in both theory and practice. In this paper, all [q + 1,2,q] and [q + 1,3,q − 1] MDS λ-constacyclic codes over GF(q) are characterized, three classes of [q + 1,4,q − 2] MDS λ-constacyclic codes over GF(q) are constructed, and four classes of [q + 1,k,q − k + 2] MDS λ-constacyclic codes over GF(q) for variable k are presented. Rationales for distinguishing the class of cyclic codes, the class of λ-constacyclic codes with λ≠ 1, and the class of non-constacyclic linear codes are given. Some applications of MDS codes in cryptography are also summarised. PubDate: 2023-01-13

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Certain simplicial complexes are used to construct a subset D of \(\mathbb {F}_{2^{n}}^{m}\) and D, in turn, defines the linear code CD over \(\mathbb {F}_{2^{n}}\) that consists of (v ⋅ d)d∈D for \(v\in \mathbb {F}_{2^{n}}^{m}\) . Here we deal with the case n = 3, that is, when CD is an octanary code. We establish a relation between CD and its binary subfield code \(C_{D}^{(2)}\) with the help of a generator matrix. For a given length and dimension, a code is called distance optimal if it has the highest possible distance. With respect to the Griesmer bound, three infinite families of distance optimal codes are obtained, and sufficient conditions for certain linear codes to be minimal are established. PubDate: 2022-12-22

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract The Galois hulls of linear codes are a generalization of the Euclidean and Hermitian hulls of linear codes. In this paper, we study the Galois hulls of (extended) GRS codes and present several new constructions of MDS codes with Galois hulls of arbitrary dimensions via (extended) GRS codes. Four general methods of constructing MDS codes with Galois hulls of arbitrary dimensions by Hermitian or general Galois self-orthogonal (extended) GRS codes are given. Using these methods, some MDS codes with larger dimensions and Galois hulls of arbitrary dimensions can be obtained. In addition, two new classes of MDS codes with explicit parameters and with Galois hulls of arbitrary dimensions are also constructed. One of the two classes can yield 1-Galois self-orthogonal GRS codes, and more new classes of MDS codes with Galois hulls of arbitrary dimensions can be derived from one of the four general methods we give. PubDate: 2022-12-17

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract In this paper, we give a general form of ideals over \(R[X]/\langle f(X)^{p^{s}}\rangle \) when f is a monic irreducible polynomial in R[X] and \(R= \mathbb {F}_{p^{m}}+u\mathbb {F}_{p^{m}}+\ldots +u^{n}\mathbb {F}_{p^{m}}\) with un+ 1 = 0. For f(X) = X − 1 this gives a general form of cyclic codes over R. Furthermore, we apply the result to classify completely the cyclic codes of length 5ps over \(\mathbb {F}_{p^{m}}+u\mathbb {F}_{p^{m}}+u^{2}\mathbb {F}_{p^{m}}\) when p ≡ 2 (mod 5) or p ≡ 3 (mod 5) and m is odd. PubDate: 2022-12-17

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Locally recoverable codes are widely used in distributed and cloud storage systems. In this paper, we first present a construction of near MDS codes with oval polynomials and then determine the locality of the codes. It turns out that the near MDS codes and their duals are both distance-optimal and dimension-optimal locally recoverable codes. The lengths of the locally recoverable codes are different from known ones in the literature. PubDate: 2022-12-05

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract In this paper, we discuss double circulant and double negacirculant codes over a family of finite rings \(R_{t} = \mathbb {F}_{q} + v_{1} \mathbb {F}_{q} + v_{2}\mathbb {F}_{q} + {\cdots } + v_{t}\mathbb {F}_{q}\) ; \(({v_{i}^{2}} = v_{i}, v_{i}v_{j} = v_{j}v_{i}=0, i,j= 1,2,\ldots ,t, i \neq j)\) , where q is an odd prime power. We obtain necessary and sufficient conditions for double circulant codes (double negacirculant codes) to be self-dual codes and to be linear codes with complementary dual (or LCD codes) codes and study the algebraic structure of self-dual and LCD double circulant (double negacirculant codes) codes. We derive a formula to determine the total number of self-dual and LCD double circulant codes (double negacirculant codes) over the ring Rt. We also find distance bounds for double circulant codes over Rt. Moreover, we use a Gray map to prove that the families of self-dual and LCD double circulant codes under this Gray map are asymptotically good. PubDate: 2022-12-03

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Abstract Multi-erasure locally recoverable codes (ME-LRCs) are an important class of locally recoverable codes (LRCs) with more constraints on the parameters. In this work, we present a new upper bound for ME-LRCs, which is tighter than that proposed by Cadambe and Mazumdar. The technique is to utilize the largest possible weight of linear codes to control the minimum distance. In addition, we present explicit constructions of ME-LRCs to achieve the proposed bound. PubDate: 2022-12-02